强网杯2024 Crypto复现

强网杯

Crypto

1. EasyRSA

题目

#encoding:utf-8
from Crypto.Util.number import long_to_bytes, bytes_to_long, getPrime
import random, gmpy2

class RSAEncryptor:
	def __init__(self):
		self.g = self.a = self.b = 0
		self.e = 65537
		self.factorGen()
		self.product()

	def factorGen(self):
		while True:
			self.g = getPrime(500)
			while not gmpy2.is_prime(2*self.g*self.a+1):
				self.a = random.randint(2**523, 2**524)
			while not gmpy2.is_prime(2*self.g*self.b+1):
				self.b = random.randint(2**523, 2**524)
			self.h = 2*self.g*self.a*self.b+self.a+self.b
			if gmpy2.is_prime(self.h):
				self.N = 2*self.h*self.g+1
				print(len(bin(self.N)))
				return

	def encrypt(self, msg):
		return gmpy2.powmod(msg, self.e, self.N)


	def product(self):
		with open('/flag', 'rb') as f:
			self.flag = f.read()
		self.enc = self.encrypt(self.flag)
		self.show()
		print(f'enc={self.enc}')

	def show(self):
		print(f"N={self.N}")
		print(f"e={self.e}")
		print(f"g={self.g}")


RSAEncryptor()

观察题目，发现是 commonprime RSA $g<a+b$ 的情况

做法一:

已知 g ,即可计算

$$h=\frac{N-1}{2g}$$ $$u=h//{2g},v=h%{2g}$$ $$u=ab+c,v=a+b+c (c是任意值)$$ $$\because \lambda(N) = 4gab$$ $$\therefore x^{4gab}\equiv 1 \pmod N$$ $$\therefore x^{4gu}=x^{4g(ab+c)}\equiv x^{4gc} \pmod N$$

设 $y=x^{4g}$
则 $y^{u}\equiv y^{c} \pmod N$
于是可以使用大步小步法试着求 dlp
对于某些$d>\sqrt{c}$,大步为

$$y^{0},y^{d},y^{2*d},\cdots,y^{d^2} \pmod N$$

小步为

$$y^{u},y^{u-1},y^{u-2},\cdots,y^{u-d} \pmod N$$

在其中搜索碰撞
存在$y^{rd}\equiv y^{u-s} \pmod N$
可以求得$c=rd+s$
从而根据 c 可以计算出 a 和 b
从而得到 p、q
exp:

from sage.all import *
from Crypto.Util.number import *
from sage.groups.generic import bsgs
from sage.all import isqrt
from sage.misc.misc_c import prod
import sage.rings.integer_ring as integer_ring
import sage.rings.integer
from sage.arith.srange import xsrange
from gmpy2 import iroot
#
# Lists of names (as strings) which the user may use to identify one
# of the standard operations:
#
multiplication_names = ('multiplication', 'times', 'product', '*')
addition_names = ('addition', 'plus', 'sum', '+')

N=24894350414989625911031283308277168570494853473112991211103425837137373907825546808317090945681095286921597859857804085402811976221748255032096758490755568701151989178807119628461722334377875332281631571643922473589681309828344656587477814419792449313365944194172112195430868664705135176694000778083946838700802246722672358404623155990644083612633942113953012749136041571421044218028388826783144812107575398236047802510961837572444973393495904391141849468050388557434387376701266993756371043482844580035711565163566803639281650857951694697419966535443017018696195483551523655526145199761912277898312419025240293141643
e=65537
g=2342316135956336269549176940731506332793030501032648498972141330132477159026231482234730908316183215946269707825148278352847015408199244188454042423299
enc=13865831661373487771647988222353027752303755915787963354245237452572394785488472237789270204907232902849008635034267057823230389802438917349808534727459429170070383709807338426183306646092021954635633251653938574360241386879982666475389204199911769664273884977317339962633706908428229504903089923038878578331925063410139538081878382857482780073530713187772451822287214686166020359278005140846983051482606015874945523091260441471317568156941474298712372990397469024231354667126351764376707763664918083507175458017675827784699241595765264379407070799046476244775502790828970579156646913503458696143725114451014880065788
nbits=2049
gamma=0.239
cbits = ceil(nbits * (0.5 - 2 * gamma))
n1=N
g1=g
v1 = ((n1 - 1) // (2*g1)) % (2*g1)
u1 = ((n1 - 1) // (2*g1)) // (2*g1)
left = (((2 * iroot(n1,2)[0]) // (2*g1)) - v1) // (2*g1)
right = (((3 * iroot(2*n1,2)[0]) // (4*g1)) - v1) // (2*g1)

dic = {}
b = pow(114514,2*g1,n1)
base = pow(b,left,n1)
D = iroot(right - left,2)[0]+1
step = pow(b,D,n1)
for i in range(D):
    dic[base] = i
    base = base * step % n1
print("baby step ready")

base = pow(b,u1,n1)
step = inverse(b,n1)
for i in range(D):
    if(base in dic):
        print("ans found!")
        print(i,dic[base])
        c = left+ i + D * dic[base]
        break
    base = base * step % n1

A = u1 - c
B = v1 + c * 2 * g1
C = iroot(B**2 - 4*A,2)[0]
x = (B+C) // 2
y = B-x
p1 = x*g1*2+1
q1 = y*g1*2+1
assert p1 * q1 == n1
print(p1)
print(q1)
phi=(p1-1)*(q1-1)
d=inverse(e,phi)
print(long_to_bytes(pow(enc,d,N)))

做法2:

爆破 a+b 的高位
${h//2g}-(a+b)_h=ab$
exp:

from tqdm import trange
from Crypto.Util.number import *
N=54642322838521966106812419124141939188989640814860878861908076164639367595914
5121966895462614693455142554411163028001396684372032321947092854090758624197340
0790290637426639994633474521237504778470452401128211738166232528044670489778765
9122983658402386409111680527237824015035597005313191262677046034069311159229487
0776292093558039681230850237745734700397178207719049329635803852591834542861388
2658054097045829584997486511884231335763642503260060763979765035367578047024986
1726402474026943128529940611865467085977350731870140237435132883515585257875892
835970695457915025407153187085139372317622088931196367733198938707
e=65537
g=30408719679598005813513822952740053880824192707932592285090992724940866129793
35548205806725469849481228948811909984262857772287453967175931780503026101
enc=169217279901286549405410484546093060498006359685672908855046065850885358778
4111525919932614689259674005902903294306990214999295401335400595465024136817616
6717262074930938187076093385743103257206008984426730319621844540768570052906856
3728141312426715117366395837964460423408187964925670546828755262865217711795726
9752597315761438459186791120018967717784674354875007341279701686250089637574512
5624115608433787391085807075889515255210830361622016035288935853286037510176320
8002340975665900236158920007589811919320440649594981349559671453007881710217298
70111273798593291304516871631013979858203217888740299565006817016165
h = (N-1)//(2*g)
# print(h)
from gmpy2 import iroot
ab_ = h//(2*g)
# print(ab - a*b)
# print((ab - a*b).bit_length())
# print(2**24-13699487)
# for i in trange(2**24):
# ab = ab_ - i
# absum = h - 2*g*ab
# if absum**2-4*ab < 0:
# continue
# abdiff = iroot(absum**2-4*ab, 2)[0]
# a = (absum + abdiff) // 2
# b = (absum - abdiff) // 2
# if a*b == ab:
# print(a)
# print(b)
# break
a = 
4793738693888445890037078687911805727587735746730025037037178739021186736608464
2223134071188443797365024207320451316918915461052607536742271505969247375076726
b = 
3081758016069703112211946605193393419929381672024605092116994783558551757717305
4601468730432823632268542332991565084729244237756343912826384891209074376593659
p = 2*g*a + 1
q = 2*g*b + 1
assert p*q == N
phi = (p-1)*(q-1)
d = pow(65537, -1, phi)
m = pow(enc,d,N)
print(long_to_bytes(m))

2. apbq

题目：

from Crypto.Util.number import *
from secrets import flag
from math import ceil
import sys

class RSA():
    def __init__(self, privatekey, publickey):
        self.p, self.q, self.d = privatekey
        self.n, self.e = publickey

    def encrypt(self, plaintext):
        if isinstance(plaintext, bytes):
            plaintext = bytes_to_long(plaintext)
        ciphertext = pow(plaintext, self.e, self.n)
        return ciphertext

    def decrypt(self, ciphertext):
        if isinstance(ciphertext, bytes):
            ciphertext = bytes_to_long(ciphertext)
        plaintext = pow(ciphertext, self.d, self.n)
        return plaintext

def get_keypair(nbits, e = 65537):
    p = getPrime(nbits//2)
    q = getPrime(nbits//2)
    n = p * q
    d = inverse(e, n - p - q + 1)
    return (p, q, d), (n, e)

if __name__ == '__main__':
    pt = './output.txt'
    fout = open(pt, 'w')
    sys.stdout = fout

    block_size = ceil(len(flag)/3)
    flag = [flag[i:i+block_size] for i in range(0, len(flag), block_size)]
    e = 65537

    print(f'[+] Welcome to my apbq game')
    # stage 1
    print(f'┃ stage 1: p + q')
    prikey1, pubkey1 = get_keypair(1024)
    RSA1 = RSA(prikey1, pubkey1)
    enc1 = RSA1.encrypt(flag[0])
    print(f'┃ hints = {prikey1[0] + prikey1[1]}')
    print(f'┃ public key = {pubkey1}')
    print(f'┃ enc1 = {enc1}')
    print(f'----------------------')

    # stage 2
    print(f'┃ stage 2: ai*p + bi*q')
    prikey2, pubkey2 = get_keypair(1024)
    RSA2 = RSA(prikey2, pubkey2)
    enc2 = RSA2.encrypt(flag[1])
    kbits = 180
    a = [getRandomNBitInteger(kbits) for i in range(100)]
    b = [getRandomNBitInteger(kbits) for i in range(100)]
    c = [a[i]*prikey2[0] + b[i]*prikey2[1] for i in range(100)]
    print(f'┃ hints = {c}')
    print(f'┃ public key = {pubkey2}')
    print(f'┃ enc2 = {enc2}')
    print(f'----------------------')

    # stage 3
    print(f'┃ stage 3: a*p + q, p + bq')
    prikey3, pubkey3 = get_keypair(1024)
    RSA3 = RSA(prikey3, pubkey3)
    enc3 = RSA2.encrypt(flag[2])
    kbits = 512
    a = getRandomNBitInteger(kbits)
    b = getRandomNBitInteger(kbits)
    c1 = a*prikey3[0] + prikey3[1]
    c2 = prikey3[0] + b*prikey3[1] 
    print(f'┃ hints = {c1, c2}')
    print(f'┃ public key = {pubkey3}')
    print(f'┃ enc3 = {enc3}')

做法:

stage1明显解方程即可

exp:

from sage.all import *
from Crypto.Util.number import *
from sympy import symbols,solve
from random import randint
from itertools import product, combinations
#stage1
hints = 18978581186415161964839647137704633944599150543420658500585655372831779670338724440572792208984183863860898382564328183868786589851370156024615630835636170
public_key = (89839084450618055007900277736741312641844770591346432583302975236097465068572445589385798822593889266430563039645335037061240101688433078717811590377686465973797658355984717210228739793741484666628342039127345855467748247485016133560729063901396973783754780048949709195334690395217112330585431653872523325589, 65537)
enc1 = 23664702267463524872340419776983638860234156620934868573173546937679196743146691156369928738109129704387312263842088573122121751421709842579634121187349747424486233111885687289480494785285701709040663052248336541918235910988178207506008430080621354232140617853327942136965075461701008744432418773880574136247
n1=public_key[0]
e1=public_key[1]
p,q=symbols('p q')
eq1=p+q-hints
eq2=p*q-n1
solutions=solve([eq1,eq2],p,q)
p1=int(solutions[0][0])
q1=int(solutions[0][1])
d1=inverse(e1,(p1-1)*(q1-1))
flag1=long_to_bytes(pow(enc1,d1,n1))
print(flag1)

stage2

做法1:

正交格
已知$(a_i,b_i)\begin{pmatrix} p \\ q \end{pmatrix} = (h_i)$
因此

$$AB_{100\times 2}\begin{pmatrix} p \\ q \end{pmatrix}=H_{100\times 1}$$

因为$a_i,b_i$都是 180bits,相对 p,q 而言很小
因此我们可以使用正交格，找到一个矩阵 M,使得

$$MH=0$$

那么,我们有

$$M \cdots AB \cdots \begin{pmatrix} p \\ q \end{pmatrix}=0$$

可以猜想,AB 即为 M 的右核经规约产生的短向量
第一步:找到这样的 M
我们可以构造一个矩阵:

$$\begin{Bmatrix} I & H \end{Bmatrix}$$

然后对 H 进行配平以使得规约出来的短向量正好满足

$$\langle X,H \rangle = 0$$

然后将所有满足这一条件的短向量合成一个矩阵
即求得 M
但实际求出的是 A-B 和 B (大小的关系)
得到 A 和 B 即可解出 p,q
exp:

from Crypto.Util.number import *
from sage.all import *
hint2 = [18167664006612887319059224902765270796893002676833140278828762753019422055112981842474960489363321381703961075777458001649580900014422118323835566872616431879801196022002065870575408411392402196289546586784096, 16949724497872153018185454805056817009306460834363366674503445555601166063612534131218872220623085757598803471712484993846679917940676468400619280027766392891909311628455506176580754986432394780968152799110962, 17047826385266266053284093678595321710571075374778544212380847321745757838236659172906205102740667602435787521984776486971187349204170431714654733175622835939702945991530565925393793706654282009524471957119991, 25276634064427324410040718861523090738559926416024529567298785602258493027431468948039474136925591721164931318119534505838854361600391921633689344957912535216611716210525197658061038020595741600369400188538567, 22620929075309280405649238349357640303875210864208854217420509497788451366132889431240039164552611575528102978024292550959541449720371571757925105918051653777519219003404406299551822163574899163183356787743543, 20448555271367430173134759139565874060609709363893002188062221232670423900235907879442989619050874172750997684986786991784813276571714171675161047891339083833557999542955021257408958367084435326315450518847393, 16581432595661532600201978812720360650490725084571756108685801024225869509874266586101665454995626158761371202939602347462284734479523136008114543823450831433459621095011515966186441038409512845483898182330730, 23279853842002415904374433039119754653403309015190065311714877060259027498282160545851169991611095505190810819508498176947439317796919177899445232931519714386295909988604042659419915482267542524373950892662544, 16542280976863346138933938786694562410542429842169310231909671810291444369775133082891329676227328401108505520149711555594236523078258701726652736438397249153484528439336008442771240980575141952222517324476607, 17054798687400834881313828738161453727952686763495185341649729764826734928113560289710721893874591843482763545781022050238655346441049269145400183941816006501187555169759754496609909352066732267489240733143973, 22115728663051324710538517987151446287208882441569930705944807337542411196476967586630373946539021184108542887796299661200933395031919501574357288914028686562763621166172668808524981253976089963176915686295217, 19324745002425971121820837859939938858204545496254632010818159347041222757835937867307372949986924646040179923481350854019113237172710522847771842257888083088958980783122775860443475680302294211764812636993025, 17269103712436870749511150569030640471982622900104490728908671745662264368118790999669887094371008536628103283985205839448583011077421205589315164079023370873380480423797655480624151812894997816254147210406492, 17365467616785968410717969747207581822018195905573214322728668902230086291926193228235744513285718494565736538060677324971757810325341657627830082292794517994668597521842723473167615388674219621483061095351780, 20823988964903136690545608569993429386847299285019716840662662829134516039366335014168034963190410379384987535117127797097185441870894097973310130525700344822429616024795354496158261293140438037100429185280939, 19068742071797863698141529586788871165176403351706021832743114499444358327620104563127248492878047796963678668578417711317317649158855864613197342671267006688211460724339403654215571839421451060657330746917459, 20089639597210347757891251257684515181178224404350699015820324544431016085980542703447257134320668961280907495580251880177990935443438799776252979843969984270461013888122703933975001704404129130156833542263882, 22344734326131457204500487243249860924828673944521980798994250859372628295695660076289343998351448667548250129358262592043131205967592613289260998148991388190917863322690137458448696392344738292233285437662495, 22688858027824961235755458925538246922604928658660170686458395195714455094516952026243659139809095639584746977271909644938258445835519951859659822660413616465736923822988993362023001205350387354001389518742538, 21286046487289796335501643195437352334100195831127922478044197411293510360710188581314023052580692810484251118253550837525637065385439859631494533102244585493243972819369812352385425700028640641292410326514111, 21542729548465815605357067072323013570796657575603676418485975214641398139843537820643982914302122976789859817102498484496409546012119998359943274203338400776158986205776474024356567247508744784200354385060666, 22319592382753357951626314613193901130171847776829835028715915533809475362288873045184870972146269975570664009921662023590318988850871708674240304838922536028975978222603171333743353770676344328056539379240160, 25195209191944761648246874631038407055240893204894145709996399690807569652160721616011712739214434932639646688187304865397816188999592774874989401871300784534538762135830014255425391132306536883804201055992313, 18257804244956449160916107602212089869395886846990320452133193087611626919926796845263727422042179229606817439442521540784268169177331707314788427670112999551683927934427716554137597798283300120796277229509678, 20293403064916574136692432190836928681820834973375054705153628740577159076332283715581047503287766236543327123639746352358718218140738999496451259789097826888955418315455420948960832865750253988992454128969953, 15967654820584966012628708475666706277218484919923639492431538068059543232562431059752700377242326527417238151501168940191488179144049286512652111172149113549072003881460743035279388672984805823560897688895124, 25144187979876039024245879200325843092774389926620026124061775431569974232758799200333888039013494603721065709195353330350750055309315207499741437181094874894647736904055829877859906318073991986020178158776286, 15736932921640444103019961538951409924080453868073105830403926861058056351553271238438325117113945341892868641345117717666354739204401152657265824568724844930574396801692131746182948347887298330990039956813130, 18831072673439732764722762485733622234889447953507582396819704359771208236721692820362137219509611319088756045211407777880521726782697895768017460064889670066178710804124631128581556314122255564861269062385337, 23800437561684813552661749774840752013501533683948618798811470214669024646396165487093720960221009038817909066075238937189371227098032581450466402462014437421254375846263830927945343485988463525070074913720710, 24402191070622494792723290726249952159888270689258801831518209605331984684494095167423722682814769395395011136124403802097229547003802312444913008194461779426175966774202219703164060353710247619639616444797670, 20215481513831963554421686543560596857659844027486522940060791775984622049024173363533378455076109165728144576719015392033536498353094895564917644840994662704362121549525329105205514332808950206092190939931448, 18384453917605955747212560280232547481041600196031285084598132475801990710125754705645482436436531608696373462641765399622296314590071558616193035939108523357020287896879479452040171765916716377102454266933226, 21890401344164908103930010123434944359446535642544335610455613014563290097498740447164765588532234051104173227090428486681237432196639010849051113283297943367655458678533223039415083212229970648958070799280218, 18379893441293694747570620009241814202936873442370354246029979042247705730610190888710981918183390028386451290137755339890329474403224043675724851314770861939082447728194632548864823398818221526652331319263027, 18715827130228986951360013590464775001019026913384718876134449689773600060962392738619405370033085704046027397895627933844824630723286144367800484157574548819065406118338665931032779491897783504790669824301288, 13588739911708699123450670852772302012518315143187739886523841133752009403411431627334135210166268158490674049617489193734568451811305631563767138879895461211915128972052001136464325219117009268526575020143259, 18506039912943821193373920483847347155611306173368341979655092778147169768984477236224526786441466933360500418090210912574990962709452725122792963919616633389125605160796446674502416801964271004625701238202575, 22167985517547342184812919437069844889650448522260359154086923601900060998572245598167213217022051141570075284051615276464952346620430587694188548679895095556459804921016744713098882496174497693878187665372865, 21507363933875318987283059841465034113263466805329282129011688531718330888226928182985538861888698160675575993935166249701145994333840516459683763957425287811252135418288516497258724668090570720893589001392220, 20250321586608105267884665929443511322540360475552916143405651419034772061789298150974629817817611591100450468070842373341756704300393352252725859102426665187194754280129749402796746118608937061141768301995522, 16104259151024766025645778755951638093681273234415510444173981198301666343334808614748361662637508091511498829253677167171091582942780017355912433497214576425697459483727777273045993446283721290714044600814203, 14560242181138184594433372530956542527312169507277535425067427080573272033961044062335960097446781943943464713852520415535775461964590009720592053626735276833191667395201287169782350381649400286337671320581068, 16239347596615402699390026749150381714807445218767496868569282767673828662340774349530405347667558555781433774705139593469838946201218537641296949822639509296966092138954685186059819628696340121356660166937131, 21344472317634795288252811327141546596291633424850284492351783921599290478005814133560171828086405152298309169077585647189366292823613547973428250604674234857289341613448177246451956695700417432794886277704716, 16053809990112020217624905718566971288375815646771826941011489252522755953750669513046736360397030033178139614200701025268874379439106827823605937814395162011464610496629969260310816473733828751702925621950679, 18917855883623050190154989683327838135081813638430345099892537186954876489710857473326920009412778140451855952622686635694323466827034373114657023892484639238914593012175120540210780102536003758794571846502397, 22690171278715056779052233972642657173540399024770527983659216197108042021644328773010698851143953503599329885607621773816718008861742027388432534850163666629476315340137626681994316866368449548292328156728206, 21087818524872480052313215092436868441694786060866149491087132591272640372512484925209820065536439188250579925233059144898601140234767300574307770064543499923712729705795392684173268461519802573563186764326797, 18439753470094841291394543396785250736332596497190578058698960152415339036714664835925822942784700917586270640813663002161425694392259981974491535370706560550540525510875465091384383255081297963169390777475352, 20105719699015744146039374208926740159952318391171137544887868739518535254000803811729763681262304539724253518465850883904308979964535242371235415049403280585133993732946919550180260852767289669076362115454200, 17251599484976651171587511011045311555402088003441531674726612079301412643514474016351608797610153172169183504289799345382527665445027976807805594288914226822374523878290416047130731166794970645275146679838899, 23027331991437585896233907022469624030630702237261170259290872847355304456043379238362120518409085840638396736666056992747627271193089116095167049248270541979716594671069985183070290375121270398623215587207529, 18158149685496169798299129683009221264185608469410295069411669832919646968324946121757411511373498747604679198739125835462814352243797919744572086307939585501566092705355693015625009717017077302201663788208609, 18276153196656501517216055049560959047263892309902154534799806637704337317207294332426798932144785240877892837491213916540255237702169595754963908689566362060228840286531616263506272071630209104758589482803348, 19830654702835464289082520892939657653574451119898587213320188332842291005863699764597454403874285715252681820027919359194554863299385911740908952649966617784376852963552276558475217168696695867402522508290055, 15349828226638644963106414986240676364822261975534684137183044733508521003843559094515387144949811552173241406076270015291925943459603622043168219534080772937297911323165839870364550841685270125556125756627553, 20923687596111161976478930953796496927811701530608223491138786355445002217973253897724452954815797952200740069102515860924306246841340715110620719064010080520601890251137419840158983682372232110885549732743013, 21095748006022412831703352650023882351218414866517568822818298949510471554885207645049385966827210564667371665855668707424105040599599901165292360321667007968065708796593851653085339928947755081203265281357013, 20136320433636422315432754195821125224777716034031656342233368000257459497472596860252592531939146543685406198978058242599116859263546329669263543660114747385041549283367183026001454445297981439938401547228229, 16496919752274418275948572022974868132658743151124597724312835413857298109100258912203517423633396955060591787380445877361136405137884456764770035346437177846666365911942996404514058688909577420388537479730705, 13788728438272498164727737074811797093818033799836159894472736480763530670013682288670889124484670336660448907074673625466218166413315342420667608074179975422284472184048790475129281850298519112884101776426380, 24852871485448795332267345793743281093931161235481251209948049584749441451621572752080662697610253315331335180611651946374137068256112152253681972406000252076016099200912670370417045090034045383991812756120791, 18663346319122078996775762643035864683521213720864038756854558668694021987970601131985163948257100423991091156649638455828855082098689641225427227191064496066436196910238564311309556938903101074363279783438714, 21400068681031931459396470039651524575262457489792894764406364952394476440804779651233022862527636114968325782197380721095406628084183336358459476006267416033892771932528688312375109463803215034905281657962293, 16044158155847172030103761204572942507195578382208455423846603003318483484698088948486132040995746837257705704187725306831142305215342467016564452582165866039427184607605673304595194959499145031211096109534167, 16518253246325822837502418827700493807621067058438396395472266350036385535241769917459657069911028720968654253735107131282350340465691670072304718987805883113410923109703284511709226857412404454224134480632696, 22032469066601123287586507039704080058983969235246539501189720236880312024198451198788699002335010120658564926677243708367430773661097221076615953342733896063909953602379936312639192315223258556134958059637605, 17474611942177808070315948910226643697957069578572244709354155010512694059987765040746148981545760660371360975936526076852619987733316042847813177383519241505024635332293992920023420060610648140841369822739716, 20097265939024591617239874622716452182434300498447992668997438018575636772416262543204370899462096267444545094719202447520254303983442269757551626971917981420832391886214473318353984504467919530676605744560570, 18170251482705061226968041449812078923477452841162650888922564215790088545936753453513162197661916172215859504545409274440450807677845894292177296835154674774694992388033874349807244020099167681146357128785394, 18084007437523118129421476751918491055914528331902780911288404344016551650138679157754567938593688369062981279371320169939281882307797009116458871503759873023914718337944953764426183937635379280572434676575757, 17001811604221128900675671565539617923973183364469396458234914432162200119518252971721448274846235879320362924206656971472493711107677598961463553324277826426691784458674010708635756004550789902368338633272118, 20217009574515126619724139485885721324936960849401637840860565569588595992087537454744066905387396266844236387315004915383456736142307523960394594650088663019228826091309049211780607761862663242437656610298243, 25534440916970201550118006203706860249111087748000550226680885431006136131742280963090650607632467666558508520152535105122661615376298673454198064361094319699307084117001019115669670029195171047304283891069792, 18871869316294018605789169171879572816494092699556970507058691345095743053290043643010965660058888064972257990750611470141816041727746767146945121588515830427165739580791663951175220638901672353681640741068573, 20173968537913641339915058056878181363456579537994317562789857397928196160113042659777558550242315788417022891612723148843142958668959046890197219991727894451795438138592005695329607326086644956073759609743066, 20601943394990265144021144365970164017319737300436518536503270346147112565303361487668388700369636611354280332841812324530501569200031186584749278453651172121161814207025650519637781007286435981682228528706305, 16397528630087028144645213166977866073543422560337716097539091258081008408890966764995645782823950721804205427713461441138000880478364026137452291234097219085473748076681729365744710225699866258812642458184750, 21373350333568141000876969785296802670776508778278005158047105058430550665787088265486222905402690421155861103648370249249790560185790723042867282734693553039477436055775198037042047438047898227097749354619822, 17767469767416052322357795736899648760868316512079849340028040817353808899589201201338152114229279980849491049574543361275046276135253417685681262008211582060955974064559129311524323185960856955462761555353091, 22148352529815091269441663541923247974004854058764556809596705832663604786920964849725772666340437231503146814919702525852955831173047034475925578238466977606367380212886384487294569287202762127531620290162734, 21663842528026621741414050256553652815372885707031383713657826718944735177083300302064509342116651731671570591336596953911570477161536730982887182434407761036442993588590230296643001682944654490645815177777455, 20219077358929317461660881724990436334639078047412693497584358963241840513748365548465302817975329987854784305275832045889690022909383530837382543579292451297269623663257098458645056099201050578472103957851128, 18255302182526662903763852563401346841065939531070045000414364747445988455597258924280193695407035356029557886165605853810182770534711966292253269625917149411889979307227493949293798772727125069093642134972336, 24926064145128749429079117171467042019887257504329103038171762786986349157515552927216574990423327013202735544601170247730647598931030432792167867343343213411600516855009788294067588153504026267213013591793027, 22369607314724468760253123915374991621544992437057652340350735935680183705467064876346663859696919167243522648029531700630202188671406298533187087292461774927340821192866797400987231509211718089237481902671100, 16994227117141934754898145294760231694287000959561775153135582047697469327393472840046006353260694322888486978811557952926229613247229990658445756595259401269267528233642142950389040647504583683489067768144570, 21758885458682118428357134100118546351270408335845311063139309657532131159530485845186953650675925931634290182806173575543561250369768935902929861898597396621656214490429009706989779345367262758413050071213624, 20156282616031755826700336845313823798147854495428660743884481573484471099887576514309769978525225369254700468742981099548840277532978306665910844928986235042420698332201264764734685502001234369189521332392642, 23291765247744127414491614915358658114280269483384022733002965612273627987872443453777028006606037159079637857473229879140366385523633075816362547967658930666106914269093225208138749470566410361196451552322613, 19807792217079652175713365065361659318870738952921195173619551645956745050506271953949139230097128034416815169649874760890189515620232505703162831090225715453502422905418824316957257395992121750661389503495033, 22074209373194902539215367382758486068533032275912313703269990627206774967653336496619231924013216321042649461711292555464574124714934511202231319963361912937842068483700298097209400217869036338644607607557860, 19678336511265998427322297909733474384702243426420286924671444552444079816707773485084891630780465895504253899943221044355971296122774264925882685351095921532685536165514189427245840338009573352081361238596378, 24746314790210393213546150322117518542380438001687269872679602687597595933350510598742749840102841364627647151669428936678130556027300886850086220074563664367409218038338623691372433831784916816798993162471163, 19346137206512895254202370018555139713690272833895195472766704715282164091959131850520571672509601848193468792313437642997923790118115476212663296111963644011010744006086847599108492279986468255445160241848708, 22739514514055088545643169404630736699361136323546717268615404574809011342622362833245601099992039789664042350284789853188040159950619203242924511038681127008964592137006103547262538912024671048254652547084347, 21491512279698208400974501713300096639215882495977078132548631606796810881149011161903684894826752520167909538856354238104288201344211604223297924253960199754326239113862002469224042442018978623149685130901455, 19381008151938129775129563507607725859173925946797075261437001349051037306091047611533900186593946739906685481456985573476863123716331923469386565432105662324849798182175616351721533048174745501978394238803081, 19965143096260141101824772370858657624912960190922708879345774507598595008331705725441057080530773097285721556537121282837594544143441953208783728710383586054502176671726097169651121269564738513585870857829805]
n, e = (73566307488763122580179867626252642940955298748752818919017828624963832700766915409125057515624347299603944790342215380220728964393071261454143348878369192979087090394858108255421841966688982884778999786076287493231499536762158941790933738200959195185310223268630105090119593363464568858268074382723204344819, 65537)
c = 30332590230153809507216298771130058954523332140754441956121305005101434036857592445870499808003492282406658682811671092885592290410570348283122359319554197485624784590315564056341976355615543224373344781813890901916269854242660708815123152440620383035798542275833361820196294814385622613621016771854846491244


L = block_matrix(ZZ,[
    [1,Matrix(ZZ,hint2).T],
])
L = L.LLL()

M = []
for i in L:
    if(i[-1] == 0):
        M.append(i[:-1])
M = Matrix(ZZ,M)

Ker = M.right_kernel().basis()
Ker = Matrix(ZZ, Ker)
AB = Ker.LLL()

B = list(map(abs, AB[1]))
A = [i+j for i,j in zip(AB[0], B)]


p = GCD(n, hint2[0]*B[1]-hint2[1]*B[0])
q = n // p
print(long_to_bytes(int(pow(c, int(inverse(e,(p-1)*(q-1))), n))))

两位学长的做法与之类似，都是找了正交格,
不过未央手搓的求正交格的轮子还是蛮有意思的

def orthogonal_lattice(B):
    _d, _n = B.nrows(), B.ncols()
    _c = 2 ** min(((_n-1)/2+(_n-_d)*(_n-_d-1)/4),20)    # this bound can be adjusted as needed
    for b in B:
        _c *= b.norm()
    B_bot = (ceil(_c)*B).stack(identity_matrix(ZZ, _n))
    B_r = B_bot.transpose().LLL()
    LB = B_r.matrix_from_rows_and_columns(range(_n-_d), range(_d,_n+_d))
    assert (B*LB.transpose()).is_zero()
    return LB

这里也附上他的 exp
exp:

from sage.all import *

v = [18167664006612887319059224902765270796893002676833140278828762753019422055112981842474960489363321381703961075777458001649580900014422118323835566872616431879801196022002065870575408411392402196289546586784096, 16949724497872153018185454805056817009306460834363366674503445555601166063612534131218872220623085757598803471712484993846679917940676468400619280027766392891909311628455506176580754986432394780968152799110962, 17047826385266266053284093678595321710571075374778544212380847321745757838236659172906205102740667602435787521984776486971187349204170431714654733175622835939702945991530565925393793706654282009524471957119991, 25276634064427324410040718861523090738559926416024529567298785602258493027431468948039474136925591721164931318119534505838854361600391921633689344957912535216611716210525197658061038020595741600369400188538567, 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21758885458682118428357134100118546351270408335845311063139309657532131159530485845186953650675925931634290182806173575543561250369768935902929861898597396621656214490429009706989779345367262758413050071213624, 20156282616031755826700336845313823798147854495428660743884481573484471099887576514309769978525225369254700468742981099548840277532978306665910844928986235042420698332201264764734685502001234369189521332392642, 23291765247744127414491614915358658114280269483384022733002965612273627987872443453777028006606037159079637857473229879140366385523633075816362547967658930666106914269093225208138749470566410361196451552322613, 19807792217079652175713365065361659318870738952921195173619551645956745050506271953949139230097128034416815169649874760890189515620232505703162831090225715453502422905418824316957257395992121750661389503495033, 22074209373194902539215367382758486068533032275912313703269990627206774967653336496619231924013216321042649461711292555464574124714934511202231319963361912937842068483700298097209400217869036338644607607557860, 19678336511265998427322297909733474384702243426420286924671444552444079816707773485084891630780465895504253899943221044355971296122774264925882685351095921532685536165514189427245840338009573352081361238596378, 24746314790210393213546150322117518542380438001687269872679602687597595933350510598742749840102841364627647151669428936678130556027300886850086220074563664367409218038338623691372433831784916816798993162471163, 19346137206512895254202370018555139713690272833895195472766704715282164091959131850520571672509601848193468792313437642997923790118115476212663296111963644011010744006086847599108492279986468255445160241848708, 22739514514055088545643169404630736699361136323546717268615404574809011342622362833245601099992039789664042350284789853188040159950619203242924511038681127008964592137006103547262538912024671048254652547084347, 21491512279698208400974501713300096639215882495977078132548631606796810881149011161903684894826752520167909538856354238104288201344211604223297924253960199754326239113862002469224042442018978623149685130901455, 19381008151938129775129563507607725859173925946797075261437001349051037306091047611533900186593946739906685481456985573476863123716331923469386565432105662324849798182175616351721533048174745501978394238803081, 19965143096260141101824772370858657624912960190922708879345774507598595008331705725441057080530773097285721556537121282837594544143441953208783728710383586054502176671726097169651121269564738513585870857829805]

def orthogonal_lattice(B):
    _d, _n = B.nrows(), B.ncols()
    _c = 2 ** min(((_n-1)/2+(_n-_d)*(_n-_d-1)/4),20)    # this bound can be adjusted as needed
    for b in B:
        _c *= b.norm()
    B_bot = (ceil(_c)*B).stack(identity_matrix(ZZ, _n))
    B_r = B_bot.transpose().LLL()
    LB = B_r.matrix_from_rows_and_columns(range(_n-_d), range(_d,_n+_d))
    assert (B*LB.transpose()).is_zero()
    return LB

from itertools import combinations

N, e = (73566307488763122580179867626252642940955298748752818919017828624963832700766915409125057515624347299603944790342215380220728964393071261454143348878369192979087090394858108255421841966688982884778999786076287493231499536762158941790933738200959195185310223268630105090119593363464568858268074382723204344819, 65537)

v = matrix(ZZ, v)
LB = orthogonal_lattice(v)
print(LB.nrows(), LB.ncols())
result = orthogonal_lattice(LB.matrix_from_rows(range(0, 100-2)))
print(result.nrows(), result.ncols())

possible_ab = []
for x, y in combinations(result, 2):
    for a_ in range(-1, 2):
        for b_ in range(-1, 2):
            z = a_*x+b_*y
            if z[0] < 0:
                possible_ab.append(-z)
            else:
                possible_ab.append(z)

print(len(possible_ab))

for a, b in combinations(possible_ab, 2):
    M = matrix(ZZ, [a, b])
    try:
        pq = M.solve_left(v)[0]
        p, q = pq[0], pq[1]
        if p * q == N:
            print(p, q)
            break
    except:
        continue 

做法2:

正常的构造关系并使用 groebner 基进行优化
随意三个 hint 可以构造以下等式:

$$a_1a_ib_2h_1-a_1a_2b_ih_1-a_1a_ib_1h_2+a_1^{2}b_ih_2+a_1a_2b_1h_i+a_1^{2}b_2h_i =0$$

提出共有系数$a_1$,可以得到

$$(a_ib_2-a_2b_i)h_1+(a_1b_i-a_ib_1)h_2+(a_2b_1+a_1b_2)h_i=0$$

得出的系数大概 360bits 左右
经过格规约出来的只是三个关系式，含有六个未知数，我们可以用 groebner 基进行约减
发现了类似$k_1a_1+k_2a_2+k_3a_3=0$的式子
然后继续 LLL 规约 $a_1,a_2,a_3$
中间有些细节需要小爆一下
最后通过 gcd 求出 p
exp:

from sage.all import *
from Crypto.Util.number import *
clist = [18167664006612887319059224902765270796893002676833140278828762753019422055112981842474960489363321381703961075777458001649580900014422118323835566872616431879801196022002065870575408411392402196289546586784096, 16949724497872153018185454805056817009306460834363366674503445555601166063612534131218872220623085757598803471712484993846679917940676468400619280027766392891909311628455506176580754986432394780968152799110962, 17047826385266266053284093678595321710571075374778544212380847321745757838236659172906205102740667602435787521984776486971187349204170431714654733175622835939702945991530565925393793706654282009524471957119991, 25276634064427324410040718861523090738559926416024529567298785602258493027431468948039474136925591721164931318119534505838854361600391921633689344957912535216611716210525197658061038020595741600369400188538567, 22620929075309280405649238349357640303875210864208854217420509497788451366132889431240039164552611575528102978024292550959541449720371571757925105918051653777519219003404406299551822163574899163183356787743543, 20448555271367430173134759139565874060609709363893002188062221232670423900235907879442989619050874172750997684986786991784813276571714171675161047891339083833557999542955021257408958367084435326315450518847393, 16581432595661532600201978812720360650490725084571756108685801024225869509874266586101665454995626158761371202939602347462284734479523136008114543823450831433459621095011515966186441038409512845483898182330730, 23279853842002415904374433039119754653403309015190065311714877060259027498282160545851169991611095505190810819508498176947439317796919177899445232931519714386295909988604042659419915482267542524373950892662544, 16542280976863346138933938786694562410542429842169310231909671810291444369775133082891329676227328401108505520149711555594236523078258701726652736438397249153484528439336008442771240980575141952222517324476607, 17054798687400834881313828738161453727952686763495185341649729764826734928113560289710721893874591843482763545781022050238655346441049269145400183941816006501187555169759754496609909352066732267489240733143973, 22115728663051324710538517987151446287208882441569930705944807337542411196476967586630373946539021184108542887796299661200933395031919501574357288914028686562763621166172668808524981253976089963176915686295217, 19324745002425971121820837859939938858204545496254632010818159347041222757835937867307372949986924646040179923481350854019113237172710522847771842257888083088958980783122775860443475680302294211764812636993025, 17269103712436870749511150569030640471982622900104490728908671745662264368118790999669887094371008536628103283985205839448583011077421205589315164079023370873380480423797655480624151812894997816254147210406492, 17365467616785968410717969747207581822018195905573214322728668902230086291926193228235744513285718494565736538060677324971757810325341657627830082292794517994668597521842723473167615388674219621483061095351780, 20823988964903136690545608569993429386847299285019716840662662829134516039366335014168034963190410379384987535117127797097185441870894097973310130525700344822429616024795354496158261293140438037100429185280939, 19068742071797863698141529586788871165176403351706021832743114499444358327620104563127248492878047796963678668578417711317317649158855864613197342671267006688211460724339403654215571839421451060657330746917459, 20089639597210347757891251257684515181178224404350699015820324544431016085980542703447257134320668961280907495580251880177990935443438799776252979843969984270461013888122703933975001704404129130156833542263882, 22344734326131457204500487243249860924828673944521980798994250859372628295695660076289343998351448667548250129358262592043131205967592613289260998148991388190917863322690137458448696392344738292233285437662495, 22688858027824961235755458925538246922604928658660170686458395195714455094516952026243659139809095639584746977271909644938258445835519951859659822660413616465736923822988993362023001205350387354001389518742538, 21286046487289796335501643195437352334100195831127922478044197411293510360710188581314023052580692810484251118253550837525637065385439859631494533102244585493243972819369812352385425700028640641292410326514111, 21542729548465815605357067072323013570796657575603676418485975214641398139843537820643982914302122976789859817102498484496409546012119998359943274203338400776158986205776474024356567247508744784200354385060666, 22319592382753357951626314613193901130171847776829835028715915533809475362288873045184870972146269975570664009921662023590318988850871708674240304838922536028975978222603171333743353770676344328056539379240160, 25195209191944761648246874631038407055240893204894145709996399690807569652160721616011712739214434932639646688187304865397816188999592774874989401871300784534538762135830014255425391132306536883804201055992313, 18257804244956449160916107602212089869395886846990320452133193087611626919926796845263727422042179229606817439442521540784268169177331707314788427670112999551683927934427716554137597798283300120796277229509678, 20293403064916574136692432190836928681820834973375054705153628740577159076332283715581047503287766236543327123639746352358718218140738999496451259789097826888955418315455420948960832865750253988992454128969953, 15967654820584966012628708475666706277218484919923639492431538068059543232562431059752700377242326527417238151501168940191488179144049286512652111172149113549072003881460743035279388672984805823560897688895124, 25144187979876039024245879200325843092774389926620026124061775431569974232758799200333888039013494603721065709195353330350750055309315207499741437181094874894647736904055829877859906318073991986020178158776286, 15736932921640444103019961538951409924080453868073105830403926861058056351553271238438325117113945341892868641345117717666354739204401152657265824568724844930574396801692131746182948347887298330990039956813130, 18831072673439732764722762485733622234889447953507582396819704359771208236721692820362137219509611319088756045211407777880521726782697895768017460064889670066178710804124631128581556314122255564861269062385337, 23800437561684813552661749774840752013501533683948618798811470214669024646396165487093720960221009038817909066075238937189371227098032581450466402462014437421254375846263830927945343485988463525070074913720710, 24402191070622494792723290726249952159888270689258801831518209605331984684494095167423722682814769395395011136124403802097229547003802312444913008194461779426175966774202219703164060353710247619639616444797670, 20215481513831963554421686543560596857659844027486522940060791775984622049024173363533378455076109165728144576719015392033536498353094895564917644840994662704362121549525329105205514332808950206092190939931448, 18384453917605955747212560280232547481041600196031285084598132475801990710125754705645482436436531608696373462641765399622296314590071558616193035939108523357020287896879479452040171765916716377102454266933226, 21890401344164908103930010123434944359446535642544335610455613014563290097498740447164765588532234051104173227090428486681237432196639010849051113283297943367655458678533223039415083212229970648958070799280218, 18379893441293694747570620009241814202936873442370354246029979042247705730610190888710981918183390028386451290137755339890329474403224043675724851314770861939082447728194632548864823398818221526652331319263027, 18715827130228986951360013590464775001019026913384718876134449689773600060962392738619405370033085704046027397895627933844824630723286144367800484157574548819065406118338665931032779491897783504790669824301288, 13588739911708699123450670852772302012518315143187739886523841133752009403411431627334135210166268158490674049617489193734568451811305631563767138879895461211915128972052001136464325219117009268526575020143259, 18506039912943821193373920483847347155611306173368341979655092778147169768984477236224526786441466933360500418090210912574990962709452725122792963919616633389125605160796446674502416801964271004625701238202575, 22167985517547342184812919437069844889650448522260359154086923601900060998572245598167213217022051141570075284051615276464952346620430587694188548679895095556459804921016744713098882496174497693878187665372865, 21507363933875318987283059841465034113263466805329282129011688531718330888226928182985538861888698160675575993935166249701145994333840516459683763957425287811252135418288516497258724668090570720893589001392220, 20250321586608105267884665929443511322540360475552916143405651419034772061789298150974629817817611591100450468070842373341756704300393352252725859102426665187194754280129749402796746118608937061141768301995522, 16104259151024766025645778755951638093681273234415510444173981198301666343334808614748361662637508091511498829253677167171091582942780017355912433497214576425697459483727777273045993446283721290714044600814203, 14560242181138184594433372530956542527312169507277535425067427080573272033961044062335960097446781943943464713852520415535775461964590009720592053626735276833191667395201287169782350381649400286337671320581068, 16239347596615402699390026749150381714807445218767496868569282767673828662340774349530405347667558555781433774705139593469838946201218537641296949822639509296966092138954685186059819628696340121356660166937131, 21344472317634795288252811327141546596291633424850284492351783921599290478005814133560171828086405152298309169077585647189366292823613547973428250604674234857289341613448177246451956695700417432794886277704716, 16053809990112020217624905718566971288375815646771826941011489252522755953750669513046736360397030033178139614200701025268874379439106827823605937814395162011464610496629969260310816473733828751702925621950679, 18917855883623050190154989683327838135081813638430345099892537186954876489710857473326920009412778140451855952622686635694323466827034373114657023892484639238914593012175120540210780102536003758794571846502397, 22690171278715056779052233972642657173540399024770527983659216197108042021644328773010698851143953503599329885607621773816718008861742027388432534850163666629476315340137626681994316866368449548292328156728206, 21087818524872480052313215092436868441694786060866149491087132591272640372512484925209820065536439188250579925233059144898601140234767300574307770064543499923712729705795392684173268461519802573563186764326797, 18439753470094841291394543396785250736332596497190578058698960152415339036714664835925822942784700917586270640813663002161425694392259981974491535370706560550540525510875465091384383255081297963169390777475352, 20105719699015744146039374208926740159952318391171137544887868739518535254000803811729763681262304539724253518465850883904308979964535242371235415049403280585133993732946919550180260852767289669076362115454200, 17251599484976651171587511011045311555402088003441531674726612079301412643514474016351608797610153172169183504289799345382527665445027976807805594288914226822374523878290416047130731166794970645275146679838899, 23027331991437585896233907022469624030630702237261170259290872847355304456043379238362120518409085840638396736666056992747627271193089116095167049248270541979716594671069985183070290375121270398623215587207529, 18158149685496169798299129683009221264185608469410295069411669832919646968324946121757411511373498747604679198739125835462814352243797919744572086307939585501566092705355693015625009717017077302201663788208609, 18276153196656501517216055049560959047263892309902154534799806637704337317207294332426798932144785240877892837491213916540255237702169595754963908689566362060228840286531616263506272071630209104758589482803348, 19830654702835464289082520892939657653574451119898587213320188332842291005863699764597454403874285715252681820027919359194554863299385911740908952649966617784376852963552276558475217168696695867402522508290055, 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21400068681031931459396470039651524575262457489792894764406364952394476440804779651233022862527636114968325782197380721095406628084183336358459476006267416033892771932528688312375109463803215034905281657962293, 16044158155847172030103761204572942507195578382208455423846603003318483484698088948486132040995746837257705704187725306831142305215342467016564452582165866039427184607605673304595194959499145031211096109534167, 16518253246325822837502418827700493807621067058438396395472266350036385535241769917459657069911028720968654253735107131282350340465691670072304718987805883113410923109703284511709226857412404454224134480632696, 22032469066601123287586507039704080058983969235246539501189720236880312024198451198788699002335010120658564926677243708367430773661097221076615953342733896063909953602379936312639192315223258556134958059637605, 17474611942177808070315948910226643697957069578572244709354155010512694059987765040746148981545760660371360975936526076852619987733316042847813177383519241505024635332293992920023420060610648140841369822739716, 20097265939024591617239874622716452182434300498447992668997438018575636772416262543204370899462096267444545094719202447520254303983442269757551626971917981420832391886214473318353984504467919530676605744560570, 18170251482705061226968041449812078923477452841162650888922564215790088545936753453513162197661916172215859504545409274440450807677845894292177296835154674774694992388033874349807244020099167681146357128785394, 18084007437523118129421476751918491055914528331902780911288404344016551650138679157754567938593688369062981279371320169939281882307797009116458871503759873023914718337944953764426183937635379280572434676575757, 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20173968537913641339915058056878181363456579537994317562789857397928196160113042659777558550242315788417022891612723148843142958668959046890197219991727894451795438138592005695329607326086644956073759609743066, 20601943394990265144021144365970164017319737300436518536503270346147112565303361487668388700369636611354280332841812324530501569200031186584749278453651172121161814207025650519637781007286435981682228528706305, 16397528630087028144645213166977866073543422560337716097539091258081008408890966764995645782823950721804205427713461441138000880478364026137452291234097219085473748076681729365744710225699866258812642458184750, 21373350333568141000876969785296802670776508778278005158047105058430550665787088265486222905402690421155861103648370249249790560185790723042867282734693553039477436055775198037042047438047898227097749354619822, 17767469767416052322357795736899648760868316512079849340028040817353808899589201201338152114229279980849491049574543361275046276135253417685681262008211582060955974064559129311524323185960856955462761555353091, 22148352529815091269441663541923247974004854058764556809596705832663604786920964849725772666340437231503146814919702525852955831173047034475925578238466977606367380212886384487294569287202762127531620290162734, 21663842528026621741414050256553652815372885707031383713657826718944735177083300302064509342116651731671570591336596953911570477161536730982887182434407761036442993588590230296643001682944654490645815177777455, 20219077358929317461660881724990436334639078047412693497584358963241840513748365548465302817975329987854784305275832045889690022909383530837382543579292451297269623663257098458645056099201050578472103957851128, 18255302182526662903763852563401346841065939531070045000414364747445988455597258924280193695407035356029557886165605853810182770534711966292253269625917149411889979307227493949293798772727125069093642134972336, 24926064145128749429079117171467042019887257504329103038171762786986349157515552927216574990423327013202735544601170247730647598931030432792167867343343213411600516855009788294067588153504026267213013591793027, 22369607314724468760253123915374991621544992437057652340350735935680183705467064876346663859696919167243522648029531700630202188671406298533187087292461774927340821192866797400987231509211718089237481902671100, 16994227117141934754898145294760231694287000959561775153135582047697469327393472840046006353260694322888486978811557952926229613247229990658445756595259401269267528233642142950389040647504583683489067768144570, 21758885458682118428357134100118546351270408335845311063139309657532131159530485845186953650675925931634290182806173575543561250369768935902929861898597396621656214490429009706989779345367262758413050071213624, 20156282616031755826700336845313823798147854495428660743884481573484471099887576514309769978525225369254700468742981099548840277532978306665910844928986235042420698332201264764734685502001234369189521332392642, 23291765247744127414491614915358658114280269483384022733002965612273627987872443453777028006606037159079637857473229879140366385523633075816362547967658930666106914269093225208138749470566410361196451552322613, 19807792217079652175713365065361659318870738952921195173619551645956745050506271953949139230097128034416815169649874760890189515620232505703162831090225715453502422905418824316957257395992121750661389503495033, 22074209373194902539215367382758486068533032275912313703269990627206774967653336496619231924013216321042649461711292555464574124714934511202231319963361912937842068483700298097209400217869036338644607607557860, 19678336511265998427322297909733474384702243426420286924671444552444079816707773485084891630780465895504253899943221044355971296122774264925882685351095921532685536165514189427245840338009573352081361238596378, 24746314790210393213546150322117518542380438001687269872679602687597595933350510598742749840102841364627647151669428936678130556027300886850086220074563664367409218038338623691372433831784916816798993162471163, 19346137206512895254202370018555139713690272833895195472766704715282164091959131850520571672509601848193468792313437642997923790118115476212663296111963644011010744006086847599108492279986468255445160241848708, 22739514514055088545643169404630736699361136323546717268615404574809011342622362833245601099992039789664042350284789853188040159950619203242924511038681127008964592137006103547262538912024671048254652547084347, 21491512279698208400974501713300096639215882495977078132548631606796810881149011161903684894826752520167909538856354238104288201344211604223297924253960199754326239113862002469224042442018978623149685130901455, 19381008151938129775129563507607725859173925946797075261437001349051037306091047611533900186593946739906685481456985573476863123716331923469386565432105662324849798182175616351721533048174745501978394238803081, 19965143096260141101824772370858657624912960190922708879345774507598595008331705725441057080530773097285721556537121282837594544143441953208783728710383586054502176671726097169651121269564738513585870857829805]
public_key2 = (73566307488763122580179867626252642940955298748752818919017828624963832700766915409125057515624347299603944790342215380220728964393071261454143348878369192979087090394858108255421841966688982884778999786076287493231499536762158941790933738200959195185310223268630105090119593363464568858268074382723204344819, 65537)
enc2 = 30332590230153809507216298771130058954523332140754441956121305005101434036857592445870499808003492282406658682811671092885592290410570348283122359319554197485624784590315564056341976355615543224373344781813890901916269854242660708815123152440620383035798542275833361820196294814385622613621016771854846491244
n2=public_key2[0]
from re import findall
from subprocess import check_output
from itertools import *


h = clist
nums = 98
L = Matrix(ZZ, 1+nums*2, 1+nums*3)
for i in range(1+nums*2):
    L[i,i] = 1
for i in range(nums):
    L[0,1+nums*2+i] = h[i+2]
    L[2*i+1,1+nums*2+i] = h[0]
    L[2*i+2,1+nums*2+i] = h[1]
L[:,-nums:] *= 2**512
L = L.LLL()

v,t,u = L[0,0],L[0,1],L[0,2]
a1s, a2s, a3s, b1s, b2s, b3s = QQ["a1,a2,a3,b1,b2,b3"].gens()
for sign in product((-1, 1), repeat=3):
    I = ideal(
        [
            a3s * b2s - a2s * b3s + sign[0] * t,
            a3s * b1s - a1s * b3s + sign[1] * u,
            a2s * b1s - a1s * b2s + sign[2] * v,
        ]
    )
    if I.dimension() != -1:
        print(sign)
        print("dim", I.dimension())
        def step2(f):
            # this f is in the form of k1*a1+k2*a2+k3*a3==0
            # for some reason, k1*b1+k2*b2+k3*b3==0 also holds
            # use LLL to find it
            print("=" * 40)
            print(f)
            L = matrix(f.coefficients()).T.augment(matrix.identity(3))
            L[:, 0] *= n2
            L = L.LLL()
            print(L[0])
            print(L[1])
            v1 = L[0]
            v2 = L[1]
            xs = []
            for c1, c2 in product((-2, -1, 0, 1, 2), repeat=2):
                v = c1 * v1 + c2 * v2
                _, x1, x2, x3 = v
                if all([0 <= x <= 2**180 for x in (x1, x2, x3)]):
                    xs.append((x1, x2, x3))
            # we don't know which one is correct pair of (a1, a2, a3) and (b1, b2, b3)
            # just try all combinations
            for g1, g2 in combinations(xs, 2):
                a1r, a2r, a3r = g1
                b1r, b2r, b3r = g2
                q = gcd(a2r * h[0] - a1r * h[1], n2)
                if 1 < q < n2:
                    p = n2 // q
                    e = 0x10001
                    d = inverse_mod(e, (p - 1) * (q - 1))
                    m = pow(enc2, d, n2)
                    flag = int(m).to_bytes(1024, "big").strip(b"\x00")
                    print(p,flag)
                    exit()

        step2(I.groebner_basis()[1])

stage3

根据题干可知

$$c_1=ap+q$$ $$c_2=p+bq$$

可以构造出如下等式

$$(c_1-p)(c_2-q)=0 \pmod n$$

展开得到

$$c_1c_2-c_1p-c_2q=0 \pmod n$$

其实这步也可以用 groenber_basis 得到同样的结果
$c_1$和$c_2$已知, p,q 接近模数的一半，于是可以小爆加 LLL 解决

$$c_1c_2-c_1(2^{brute}p_h+i)-c_2(2^{brute}q_h+j)=0 \pmod n$$

exp:

from Crypto.Util.number import *
from sage.all import *
#stage3
hint3 = (68510878638370415044742935889020774276546916983689799210290582093686515377232591362560941306242501220803210859757512468762736941602749345887425082831572206675493389611203432014126644550502117937044804472954180498370676609819898980996282130652627551615825721459553747074503843556784456297882411861526590080037, 117882651978564762717266768251008799169262849451887398128580060795377656792158234083843539818050019451797822782621312362313232759168181582387488893534974006037142066091872636582259199644094998729866484138566711846974126209431468102938252566414322631620261045488855395390985797791782549179665864885691057222752)
n3,e3 = (94789409892878223843496496113047481402435455468813255092840207010463661854593919772268992045955100688692872116996741724352837555794276141314552518390800907711192442516993891316013640874154318671978150702691578926912235318405120588096104222702992868492960182477857526176600665556671704106974346372234964363581, 65537)
enc3 = 17737974772490835017139672507261082238806983528533357501033270577311227414618940490226102450232473366793815933753927943027643033829459416623683596533955075569578787574561297243060958714055785089716571943663350360324047532058597960949979894090400134473940587235634842078030727691627400903239810993936770281755
c1,c2 = hint3

brute = 2
for i in range(2^brute):
    for j in range(2^brute):
        L = Matrix(ZZ, [
            [1,0,0,2^brute*c1],
            [0,1,0,2^brute*c2],
            [0,0,2^(512-brute),c1*i+c2*j-c1*c2],
            [0,0,0,n3]
        ])
        L[:,-1:] *= n3
        res = L.LLL()[0]

        p = 2^brute*abs(res[0])+i
        if(n3 % p == 0):
            print(p)

exp:

from sage.all import * 
from Crypto.Util.number import *
from sympy import symbols,solve
from random import randint
from itertools import product, combinations
#stage1
hints = 18978581186415161964839647137704633944599150543420658500585655372831779670338724440572792208984183863860898382564328183868786589851370156024615630835636170
public_key = (89839084450618055007900277736741312641844770591346432583302975236097465068572445589385798822593889266430563039645335037061240101688433078717811590377686465973797658355984717210228739793741484666628342039127345855467748247485016133560729063901396973783754780048949709195334690395217112330585431653872523325589, 65537)
enc1 = 23664702267463524872340419776983638860234156620934868573173546937679196743146691156369928738109129704387312263842088573122121751421709842579634121187349747424486233111885687289480494785285701709040663052248336541918235910988178207506008430080621354232140617853327942136965075461701008744432418773880574136247
n1=public_key[0]
e1=public_key[1]
p,q=symbols('p q')
eq1=p+q-hints
eq2=p*q-n1
solutions=solve([eq1,eq2],p,q)
p1=int(solutions[0][0])
q1=int(solutions[0][1])
d1=inverse(e1,(p1-1)*(q1-1))
flag1=long_to_bytes(pow(enc1,d1,n1))
print(flag1)
#stage2
hints2 = [18167664006612887319059224902765270796893002676833140278828762753019422055112981842474960489363321381703961075777458001649580900014422118323835566872616431879801196022002065870575408411392402196289546586784096, 16949724497872153018185454805056817009306460834363366674503445555601166063612534131218872220623085757598803471712484993846679917940676468400619280027766392891909311628455506176580754986432394780968152799110962, 17047826385266266053284093678595321710571075374778544212380847321745757838236659172906205102740667602435787521984776486971187349204170431714654733175622835939702945991530565925393793706654282009524471957119991, 25276634064427324410040718861523090738559926416024529567298785602258493027431468948039474136925591721164931318119534505838854361600391921633689344957912535216611716210525197658061038020595741600369400188538567, 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22074209373194902539215367382758486068533032275912313703269990627206774967653336496619231924013216321042649461711292555464574124714934511202231319963361912937842068483700298097209400217869036338644607607557860, 19678336511265998427322297909733474384702243426420286924671444552444079816707773485084891630780465895504253899943221044355971296122774264925882685351095921532685536165514189427245840338009573352081361238596378, 24746314790210393213546150322117518542380438001687269872679602687597595933350510598742749840102841364627647151669428936678130556027300886850086220074563664367409218038338623691372433831784916816798993162471163, 19346137206512895254202370018555139713690272833895195472766704715282164091959131850520571672509601848193468792313437642997923790118115476212663296111963644011010744006086847599108492279986468255445160241848708, 22739514514055088545643169404630736699361136323546717268615404574809011342622362833245601099992039789664042350284789853188040159950619203242924511038681127008964592137006103547262538912024671048254652547084347, 21491512279698208400974501713300096639215882495977078132548631606796810881149011161903684894826752520167909538856354238104288201344211604223297924253960199754326239113862002469224042442018978623149685130901455, 19381008151938129775129563507607725859173925946797075261437001349051037306091047611533900186593946739906685481456985573476863123716331923469386565432105662324849798182175616351721533048174745501978394238803081, 19965143096260141101824772370858657624912960190922708879345774507598595008331705725441057080530773097285721556537121282837594544143441953208783728710383586054502176671726097169651121269564738513585870857829805]
public_key2 = (73566307488763122580179867626252642940955298748752818919017828624963832700766915409125057515624347299603944790342215380220728964393071261454143348878369192979087090394858108255421841966688982884778999786076287493231499536762158941790933738200959195185310223268630105090119593363464568858268074382723204344819, 65537)
enc2 = 30332590230153809507216298771130058954523332140754441956121305005101434036857592445870499808003492282406658682811671092885592290410570348283122359319554197485624784590315564056341976355615543224373344781813890901916269854242660708815123152440620383035798542275833361820196294814385622613621016771854846491244
h1=hints2[23]
h2=hints2[1]
h3=hints2[5]
h4=hints2[33]
n=public_key2[0]
e=public_key2[1]
import itertools
hints=hints2[0:4]
V = hints
k = 2**800
M = Matrix.column([k * v for v in V]).augment(Matrix.identity(len(V)))
B = [b[1:] for b in M.LLL()]
M = (k * Matrix(B[:len(V)-2])).T.augment(Matrix.identity(len(V)))
B = [b[-len(V):] for b in M.LLL() if set(b[:len(V)-2]) == {0}]
p1=735663074887631225801798676262526429409552987487528189190178286249638327007669154091250575156243472996039447903422153802207289643930712614541433488783691929790870903948581
print(p1.bit_length())
for s, t in itertools.product(range(4), repeat=2):
    T = s*B[0] + t*B[1]
    a1, a2, a3,a4 = T
    kq = gcd(a1 * hints[1] - a2 * hints[0], n)
    if 1 < kq < n:
        print('find!', kq, s, t)
        break
for i in range(2**16, 1, -1):
    if kq % i == 0:
        kq //= i
q = int(kq)
p = int(n // kq)
print(p)
print(q)
d = pow(0x10001, -1, (p - 1) * (q - 1))
m = pow(enc2, d, n)
flag2= long_to_bytes(m)

p=8112940945910485817171807897687451701452029959677470272197529542411816926233172848066074195817612280582244564398252967013953964546888998662975298338523549
q=9067773077510925207378520309595658022345214442920360440202890774224295250116442048990578009377300541280465330975931465993745130297479191298485033569345231
enc3 = 17737974772490835017139672507261082238806983528533357501033270577311227414618940490226102450232473366793815933753927943027643033829459416623683596533955075569578787574561297243060958714055785089716571943663350360324047532058597960949979894090400134473940587235634842078030727691627400903239810993936770281755
flag3=long_to_bytes(pow(enc3,d,n))
flag=flag1+flag2+flag3
print(flag)
'''
load('https://gist.githubusercontent.com/Connor-McCartney/952583ecac836f843f50b785c7cb283d/raw/5718ebd8c9b4f9a549746094877a97e7796752eb/solvelinmod.py')
hints3 = (68510878638370415044742935889020774276546916983689799210290582093686515377232591362560941306242501220803210859757512468762736941602749345887425082831572206675493389611203432014126644550502117937044804472954180498370676609819898980996282130652627551615825721459553747074503843556784456297882411861526590080037, 117882651978564762717266768251008799169262849451887398128580060795377656792158234083843539818050019451797822782621312362313232759168181582387488893534974006037142066091872636582259199644094998729866484138566711846974126209431468102938252566414322631620261045488855395390985797791782549179665864885691057222752)
public_key3 = (94789409892878223843496496113047481402435455468813255092840207010463661854593919772268992045955100688692872116996741724352837555794276141314552518390800907711192442516993891316013640874154318671978150702691578926912235318405120588096104222702992868492960182477857526176600665556671704106974346372234964363581, 65537)
enc3 = 17737974772490835017139672507261082238806983528533357501033270577311227414618940490226102450232473366793815933753927943027643033829459416623683596533955075569578787574561297243060958714055785089716571943663350360324047532058597960949979894090400134473940587235634842078030727691627400903239810993936770281755
n=public_key2[0]
h1=hints3[1]
h2=hints3[0]
var('p q')
bounds = {p: 2**512, q: 2**512}
eqs = [(q*h1 + p*h2 - h1*h2==0, n)]
sol = solve_linear_mod(eqs, bounds)
p = sol[p]
q = sol[q]
print("p=",p)
print("q=",q)
assert p*q==n
d = inverse(65537,(p-1)*(q-1))
print(long_to_bytes(pow(enc3,int(d),int(n))))
'''

'''
M = Matrix([
    [h2,h4,0,0,1,0,0,0],
    [-h1,0,h3,0,0,1,0,0],
    [0,0,-h2,h4,0,0,1,0],
    [0,-h1,0,-h3,0,0,0,1],
    [n2,0,0,0,0,0,0,0],
    [0,n2,0,0,0,0,0,0],
    [0,0,n2,0,0,0,0,0],
    [0,0,0,n2,0,0,0,0]
])
k = 2**(512+180)
W = diagonal_matrix([k, k, k,k,1, 1, 1, 1])
M = (M*W).LLL() / W
for row in M:
    
    if row[:4] != 0:
        continue
    _, _, _,_, x1, x2, x3 ,x4 = row
    
    for x in (x1, x2, x3,x4):
        
        p2 = gcd(int(abs(x)),n2)
        
        if p2 == 1:
            continue
        q2 = n2//p2
        d2 = pow(e2, -1, (p2-1)*(q2-1))
        print(long_to_bytes(int(pow(enc2, d2, n2))))
'''        

3. 21_step

题目:

import re
import random
from secrets import flag
print(f'Can you weight a 128 bits number in 21 steps')
pattern = r'([AB]|\d+)=([AB]|\d+)(\+|\-|\*|//|<<|>>|&|\^|%)([AB]|\d+)'

command = input().strip()
assert command[-1] == ';'
assert all([re.fullmatch(pattern, i) for i in command[:-1].split(';')])

step = 21
for i in command[:-1].split(';'):
    t = i.translate(str.maketrans('', '', '=AB0123456789'))
    if t in ['>>', '<<', '+', '-', '&', '^']:
        step -= 1
    elif t in ['*', '/', '%']:
        step -= 3
if step < 0:exit()

success = 0
w = lambda x: sum([int(i) for i in list(bin(x)[2:])])
for _ in range(100):
    A = random.randrange(0, 2**128)
    wa = w(A)
    B = 0
    try : exec("global A; global B;" + command)
    except : exit()
    if A == wa:
        success += 1

if success == 100:
    print(flag)

做法:

要在21步之内测出一个数的权重,可以使用汉明权重算法
原理:
这里是 优化的SWAR 算法
先采用分治思想
优化前(朴素的 SWAR 算法)
即先两个两个一组，计算出每两个含1的个数
然后再四个四个、八个八个、以此类推
每一步运算可以进行优化
以下是优化之后的代码

A=A-((A>>1)&0x55555555555555555555555555555555)
A=(A&0x33333333333333333333333333333333)+((A>>2)&0x33333333333333333333333333333333)
A=((A+(A>>4))&0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f)
A=A+(A>>8)
A=A+(A>>16)
A=A+(A>>32)
A=A+(A>>64)
A=A&127

exp:

from pwn import *
command = '''B=A>>1;B=B&113427455640312821154458202477256070485;B=A-B;A=B&68056473384187692692674921486353642291;B=B>>2;B=B&68056473384187692692674921486353642291;B=B+A;A=B>>4;B=B+A;B=B&20016609818878733144904388672456953615;A=B>>8;B=B+A;A=B>>16;B=B+A;A=B>>32;B=B+A;A=B>>64;B=B+A;B=B&127;A=B+0;'''
io=remote("47.93.15.136",37092)
io.recvline()
#io.send(command)
io.interactive()
#flag{you_can_weight_it_in_21_steps!}

4.traditional_game

题目:

from Crypto.Util.number import getPrime,bytes_to_long
from secret import secret,flag
import random
import time
import os
import signal

def _handle_timeout(signum, frame):
    raise TimeoutError('function timeout')

timeout = 300
signal.signal(signal.SIGALRM, _handle_timeout)
signal.alarm(timeout)

random.seed(secret + str(int(time.time())).encode())

class RSA:
    def __init__(self):
        self.p = getPrime(512)
        self.q = getPrime(512)
        self.e = getPrime(128)
        self.n = self.p * self.q
        self.phi = (self.p - 1) * (self.q - 1)
        self.d = pow(self.e, -1, self.phi)  

    def get_public_key(self):
        return (self.n, self.e)
    
    def get_private_key(self, blind_bit=None, unknown_bit=None):
        if blind_bit is not None and unknown_bit is not None:
            blind = getPrime(blind_bit)
            d_ = ((int(self.d >> unknown_bit) // blind * blind) << unknown_bit) + int(self.d % blind)
            return (d_, blind)
        else:
            return (self.d, 0)
    
    def encrypt(self, m):
        if type(m) == bytes:
            m = bytes_to_long(m)
        elif type(m) == str:
            m = bytes_to_long(m.encode())
        return pow(m, self.e, self.n)
    
    def game(self,m0,m1,b):   
        return self.encrypt([m0,m1][b]) 


rsa = RSA()
token = os.urandom(66) 

print( "[+] Welcome to the game!")
print(f"[+] rsa public key: {rsa.get_public_key()}")

coins = 100
price = 100
while coins > 0:
    print("=================================")
    b = random.randint(0,1)
    c = rsa.game(
        b'bit 0:' + os.urandom(114), 
        b'bit 1:' + os.urandom(114), 
        b)
    print("[+] c:",c)
    guessb = int(input("[-] b:"))
    coins -= 1
    if guessb == b:
        price -= 1
        print("[+] correct!") 
    else: 
        print("[+] wrong!") 

if price != 0: 
    print("[-] game over!")
    exit()

blind_bit = 40
unknown_bit = 365

d_,blind = rsa.get_private_key(blind_bit, unknown_bit)

print( "[+] Now, you have permission to access the privkey!")
print(f"[+] privkey is: ({d_},{blind}).")
print(f"[+] encrypt token is: {rsa.encrypt(bytes_to_long(token))}")

guess_token = bytes.fromhex(input("[-] guess token:"))
if guess_token == token:
    print("[+] correct token, here is your flag:",flag)
else:
    print("[-] wrong token")

做法:

第一关:
侧信道,seed与时间有关，于是可以同时连两个靶机，一个看答案，一个填答案。
第二关:
私钥部分泄露
首先复现鸡块师傅的wp

dh = (d_ >> unknown_bit) // blind
dl = d_ % blind

将 blind 记为 B,可以得到如下式子：

$$d = 2^{365}Bd_h+Bd_m+d_l$$

其中$d_m$未知
有了 d的近似高位和e
由$ed=k \phi(n)+1$得

$$ed_h \approx kn$$

于是可以求出 d 的近似值
继续代入等式$ed=k \phi(n)+1$
将已知数据代入,可得

$$e(2^{365}Bd_h+Bd_m+d_l$)=1+k(n-p-q+1)$$

整理得

$$e(2^{365}Bd_h+d_l)-k(n+1)+eBd_m+k(p+q)-1=0$$

设$M=e(2^{365}Bd_h+d_l)-k(n+1)-1$
于是有

$$eBd_m+k(p+q)=0 \pmod M$$

$M$ 比 $d_m$ 和 $(p+q)$都要大，所以可以使用 LLL 进行规约求解
但是 LLL 并没有得到预期结果，而是找到了一个更短的向量
这个向量为

$$\pm (-k,eB,0)$$

而第二行为

$$(d_m,p+q,0)-t(-k,eB,0)$$

因为第二行数量级与$p+q$的数量级相近,于是可以得到$p+q$的高位
$(p+q)%eB$即为$p+q$的低位
于是可以转化为copper问题
将已知p+q高低位问题转化为已知p高低位问题
(低位是模eB意义下的)
首先在精度环上找到p的高位

#sage
PR.<x> = PolynomialRing(RealField(1000))
f = x*(pqh-x) - n 
ph = int(f.roots()[0][0]) // (e*blind) * (e*blind)

其次分别在模 e 和 B 意义下去寻找 p的低位,然后做 CRT,求得 p 模 eB 意义下的低位

#sage
R.<x> = PolynomialRing(Zmod(e))
fe = x*(pql-x) - n
rese = fe.roots()

R.<x> = PolynomialRing(Zmod(blind))
fb = x*(pql-x) - n
resb = fb.roots()

for i in rese:
    for j in tqdm(resb):
        nlist = [e,blind]
        clist = [int(i[0]),int(j[0])]
        pl = int(crt(clist,nlist))

然后就成功转化为已知p高低位的coppersmith

$$p=p_h+eB*x+p_l$$

然后对该式在模 n 意义下求小根即可(不过调参我是真调不明白，回头系统学一下)
自己重写的exp:
exp:

from sage.all import *
from Crypto.Util.number import *
#set_verbose(2)
n, e = 85326799303014496026064371730108772674311332235197622668632563161991602151711567135121280447576711444698674128128361298108919951648223986067215747231280671858955299505770967232491254103263536714363392698361887554304255293991522369123049450389169430042203506125207632731477059899911948136482137033584148217327, 254105853966712863308295955275449077351
d_, blind = 21378651607895524021279554589638483737491146093477974150650211288463988209605132503571807152155799637952833578907915487652394101057156932695702372344185513500134378667094923442647974423915643213700386663283702204046537471425141802799603607927835879677131664511051343930330150649244588933823965535065854996174,1033378191583
c = 53742732129028328063112055299975805421399396945032126712022723372351554830940247890238890674346304685771599765645061449905866414886468887177488826724343823941484173305683401484367061591100780027467534437239106132736115748345075082986831034406103970493456683245813494600511020286115561513979129441895596164513
blind_bit = 40
unknown_bit = 365
k = (e*d_ - 1) // n + 1
dh = (d_ >> (365)) // blind
dl = d_ % blind


M=e*(2**365*blind*dh+dl)-k*(n+1)-1
K=2**(513-unknown_bit)
L=matrix(ZZ,[[K,0,e*blind],[0,1,k],[0,0,M]])
L[:,-1:] *= M
L = L.LLL()
L[:,:1] /= K
#print(L)
#print(int(L[1][1]).bit_length())
pqh = L[1][1] // (e*blind) * (e*blind)
pql = L[1][1] % (e*blind)
#print(pqh)
#pqh=18507769095935379124387654959898265148693230444521588629361097006302149784602492419374694127887114275959323662548976532766729044254524730977398946311925444
PR=PolynomialRing(RealField(1000),'x')
x=PR.gen()
f = x*(pqh-x) - n 
ph = int(f.roots()[0][0])//(e*blind)*(e*blind)
#print(ph)
R=PolynomialRing(Zmod(e),'x')
x=R.gen()
f=x*(pql-x)-n
re=f.roots()
R=PolynomialRing(Zmod(blind),'x')
x=R.gen()
f=x*(pql-x)-n
rb=f.roots()
n, e = 85326799303014496026064371730108772674311332235197622668632563161991602151711567135121280447576711444698674128128361298108919951648223986067215747231280671858955299505770967232491254103263536714363392698361887554304255293991522369123049450389169430042203506125207632731477059899911948136482137033584148217327, 254105853966712863308295955275449077351
d_, blind = 21378651607895524021279554589638483737491146093477974150650211288463988209605132503571807152155799637952833578907915487652394101057156932695702372344185513500134378667094923442647974423915643213700386663283702204046537471425141802799603607927835879677131664511051343930330150649244588933823965535065854996174,1033378191583
c = 53742732129028328063112055299975805421399396945032126712022723372351554830940247890238890674346304685771599765645061449905866414886468887177488826724343823941484173305683401484367061591100780027467534437239106132736115748345075082986831034406103970493456683245813494600511020286115561513979129441895596164513
ph= 8699285662194670581068573248270824005815358225863866569197658526663593487792213714033312321399166457288086214766777389663713682155573983780505533662370967
rese= [(62152471902631138505624687104331817667, 1), (8435944108528164915116898250995780984, 1)]
resb= [(727798351633, 1), (588657551195, 1)]
for i in re:
    for j in rb:
        nlist=[e,blind]
        clist=[int(i[0]),int(j[0])]
        pl=int(crt(clist,nlist))
        PR=PolynomialRing(Zmod(n),'x')
        x=PR.gen()
        f=ph+e*blind*x+pl 
        root=f.monic().small_roots(X=2**(512-105-167+3),beta=0.499,epsilon=0.02)
        print(root)
        if root!=[]:
            p=int(ph+e*blind*root[0]+pl)
            q=n//p 
            d=inverse(e,(p-1)*(q-1))
            print(long_to_bytes(pow(c,d,n)).hex())  

5. ECRandom_game

题目:

from flag import M, q, a, b, select
import hashlib
from hashlib import sha256
from Crypto.Util.number import *
from Crypto.Cipher import AES
import sys
import ecdsa
from Crypto.Util.Padding import pad, unpad
from ecdsa.ellipticcurve import CurveFp,Point
from math import ceil
import os
import random
import string

flag = b'qwb{kLeMjJw_HBPtoHsVhnnxZdvtGjomivNDUI_vMRhZHrfKlCZ6HlGAeXRV_gQ8i117nGhzEMr0Zk_YTl1wftSskpX4JLnryE9Mhl96cPTWorGCl_R6nD33bcx1AYflag_leak}'
assert len(flag) == 136

BANNER = '''
 GGGGG   OOOOO   DDDDD   DDDDD    GGGGG    AAA    MM    MM EEEEEEE 
GG      OO   OO  DD   D  DD   D  GG       AAAAA   MMM  MMM EE      
GG  GGG OO   OO  DD   D  DD   D  GG  GGG  A   A   MM MM MM EEEEE   
GG   GG OO   OO  DD   D  DD   D  GG   GG  AAAAA   MM    MM EE      
GGGGGGG  OOOOO   DDDDD   DDDDD   GGGGGGG  A   A   MM    MM EEEEEEE 
 '''

NNN = []
def die(*args):
    pr(*args)
    quit()

def pr(*args):
    s = " ".join(map(str, args))
    sys.stdout.write(s + "\n")
    sys.stdout.flush()

def sc():
    return sys.stdin.buffer.readline()

def Rng(k):
    ran = random.getrandbits(k)
    NNN.append(ran)
    return ran


def ADD(num):
    NUM = 0
    for i in range(num):
        NUM += Rng(32)
    return NUM

def secure_choice(sequence):
    if not sequence:
        return None
    randbelow = 0
    for i, x in enumerate(sequence):
        randbelow += 1
        if os.urandom(1)[0] < (1 << 8) * randbelow // len(sequence):
            return x

def xDBLADD(P, Q, PQ, q, a, b):
    (X1, Z1), (X2, Z2), (X3, Z3) = PQ, P, Q
    X4 = (X2**2 - a * Z2**2) ** 2 - 8 * b * X2 * Z2**3
    Z4 = 4 * (X2 * Z2 * (X2**2 + a * Z2**2) + b * Z2**4)
    X5 = Z1 * ((X2 * X3 - a * Z2 * Z3) ** 2 - 4 * b * Z2 * Z3 * (X2 * Z3 + X3 * Z2))
    Z5 = X1 * (X2 * Z3 - X3 * Z2) ** 2
    X4, Z4, X5, Z5 = (c % q for c in (X4, Z4, X5, Z5))
    return (X4, Z4), (X5, Z5)

def xMUL(P, k, q, a, b):
    Q, R = (1, 0), P
    for i in reversed(range(k.bit_length() + 1)):
        if k >> i & 1:
            R, Q = Q, R
        Q, R = xDBLADD(Q, R, P, q, a, b)
        if k >> i & 1:
            R, Q = Q, R
    return Q

def shout(x, d, q, a, b):
    P = (x,1)
    Q = xMUL(P, d, q, a, b)
    return Q[0] * pow(Q[1], -1, q) % q

def generate_random_string(length):
    characters = string.ascii_letters + string.digits
    random_string = ''.join(secure_choice(characters) for i in range(length))
    return random_string

class ECCDu():
    def __init__(self,curve,G):
        self.state = getRandomNBitInteger(512)
        self.Curve = curve
        self.g = G

        self.num = Rng(32)
        d = 65537
        self.Q = self.num*self.g
        self.P = d * self.Q

    def ingetkey(self):
        t = int((self.state * self.Q).x())
        self.updade()
        return t%(2**250)

    def updade(self):
        self.state = int((self.state * self.P).x())

    def Random_key(self, n:int):
        out = 0
        number = ceil(n/250)
        for i in range(number):
            out = (out<<250) + self.ingetkey()
        return out % (2**n)

def proof_of_work():
    random.seed(os.urandom(8))
    proof = ''.join([random.choice(string.ascii_letters + string.digits) for _ in range(20)])
    _hexdigest = sha256(proof.encode()).hexdigest()
    pr(f"sha256(XXXX+{proof[4:]}) == {_hexdigest}".encode())
    pr('Give me XXXX: ')
    x = sc().rstrip(b'\n')
    if len(x) != 4 or sha256(x + proof[4:].encode()).hexdigest() != _hexdigest:
        return False
    return True

def main():
    pr(BANNER)
    pr('WELCOME TO THIS SIMPLE GAME!!!')
    ASSERT = proof_of_work()
    if not ASSERT:
        die("Not right proof of work")

    pr('Now we will start our formal GAME!!!')
    pr('===== First 1💪: =====')
    pr('Enter an integer as the parameter p for Curve: y^2 = x^3+12x+17 (mod p) and 250<p.bit_length()')
    p1 = int(sc())
    if not 250<=p1.bit_length():
        die('Wrong length!')
    curve = CurveFp(p1, 12, 17,1)
    pr(curve)
    pr('Please Enter a random_point G:')
    G_t = sc().split(b' ')
    Gx,Gy = int(G_t[0]),int(G_t[1])
    if not curve.contains_point(Gx,Gy):
        die('This point is outside the curve')
    G = Point(curve,Gx,Gy)

    for i in range(500):
        ECDU = ECCDu(curve,G)
        m  = 'My secret is a random saying of phrase,As below :' + generate_random_string(119)
        Number = ECDU.Random_key(1344)
        c = Number^bytes_to_long(m.encode())
        pr(f'c = {c}')
        pr(f'P = {int(ECDU.P.x()), int(ECDU.P.y())}')
        pr(f'Q = {int(ECDU.Q.x()), int(ECDU.Q.y())}')

        pr('Enter m:')
        m_en = sc().rstrip(b'\n')
        if m_en != m.encode():
            die('This is not the right m,Please try again')
        else:
            pr('Right m!!!')
    pr('Bingo!')
    new_state,new_state1 = ADD(136),ADD(50)
    
    random.seed(new_state)

    pr('===== Second 2💪: =====')
    curve1 = CurveFp(q,a,b,1)
    pr(f'{int(curve1.p()),int(curve1.a()),int(curve1.b())}')

    pr("Enter a number that does not exceed 1500")
    number = int(sc())

    pr(f'You only have {number} chances to try')
    success = 0
    for i in range(number):
        Gx = secure_choice(select)
        R = Rng(25)
        pr('Gx = ',Gx)
        pr('Px = ',shout(Gx, R, q, a, b))
        R_n = int(sc().rstrip(b'\n'))
        if R_n != R:
            pr(f'Wrong number!!!,Here is your right number {R}')
        else:
            pr('GGood!')
            success += 1

    if int(success)/int(number) <= 0.262:
        die('Please Try more...')

    random.seed(new_state)
    iv_num = ADD(2000)

    iv = hashlib.sha256(str(iv_num).encode()).digest()[:16]
    key = hashlib.sha256(str(new_state1).encode()).digest()[:16]
    Aes = AES.new(key,AES.MODE_CBC, iv)
    C = Aes.encrypt(pad(flag,16))

    key_n, iv_n = int(sc()), int(sc())
    iv1 = hashlib.sha256(str(iv_n).encode()).digest()[:16]
    key1 = hashlib.sha256(str(key_n).encode()).digest()[:16]
    Aes_verify = AES.new(key1,AES.MODE_CBC, iv1)
    C_verify = unpad(Aes_verify.decrypt(C),16)

    if C_verify == flag:
        pr('Congratulations🎇🎇🎇!!!!')
        pr('Here is your flag😊：', M)
    else:
        pr('Maybe you could get the right flag!')

if __name__ == '__main__':
    main()

做法:

step1:
还是老经典了，proof_of_work,爆一下工作量证明即可

step2:
总共 500 次，每次生成了一个 key,但根据其分块方式高$1500-1344$位是无效位，可以不用管

其余依次分为$94 | 250 | 250 | 250 | 250 | 250$

因为已知的前缀明文有49个字符。也就是392位泄露
因此第二块是完全知道的,第三块知道$392-94-250=48$也就是高48位
然后由于素数 p 可以由我们控制，就可以让它略大于250位,使得$t%2^{250}=t$
由第二块的 t1 可以通过计算得到第三块的 state ,从而递推得到所有的 state
得到所有的 state 后就可以得到 key 从而过关

step3:
500 轮之后,$ADD(136)$和$ADD(50)$分别生成了136个和50个$Rng(32)$之和
因为MT19937具有周期性,拥有 624 个连续生成的随机数就可以预测所有随机数，因此 new_state1 是完全可以预测的
审计后面的代码，可知现在得到的 new_state1 即是后面 AES 的 key

step4:
已知椭圆曲线方程还有 P 和 G ,一眼看去是个 25bit 的数的 ECDLP
实际上不用进行 ECDLP ,因为没有限制最大是1500,所以我们可以通过报错拿到数据,然后再全部预测成功后面的数据即可
预测完之后求到 seed 然后再发送即可
复现exp:

from sage.all import *
import hashlib
from hashlib import sha256
from Crypto.Util.number import *
from Crypto.Cipher import AES
import sys
import ecdsa
from Crypto.Util.Padding import pad, unpad
from ecdsa.ellipticcurve import CurveFp,Point
from math import ceil
import os
import random
import string
from gmpy2 import invert

M=b"flag{You_have_defeated_MT19937_and_ECC!!!}"
a=getRandomInteger(256)
b=getRandomInteger(256)
q=getPrime(256)
flag = b'qwb{kLeMjJw_HBPtoHsVhnnxZdvtGjomivNDUI_vMRhZHrfKlCZ6HlGAeXRV_gQ8i117nGhzEMr0Zk_YTl1wftSskpX4JLnryE9Mhl96cPTWorGCl_R6nD33bcx1AYflag_leak}'
assert len(flag) == 136
select=os.urandom(16)
BANNER = '''
 GGGGG   OOOOO   DDDDD   DDDDD    GGGGG    AAA    MM    MM EEEEEEE 
GG      OO   OO  DD   D  DD   D  GG       AAAAA   MMM  MMM EE      
GG  GGG OO   OO  DD   D  DD   D  GG  GGG  A   A   MM MM MM EEEEE   
GG   GG OO   OO  DD   D  DD   D  GG   GG  AAAAA   MM    MM EE      
GGGGGGG  OOOOO   DDDDD   DDDDD   GGGGGGG  A   A   MM    MM EEEEEEE 
 '''

NNN = []
def die(*args):
    pr(*args)
    quit()

def pr(*args):
    s = " ".join(map(str, args))
    sys.stdout.write(s + "\n")
    sys.stdout.flush()

def sc():
    return sys.stdin.buffer.readline()

def Rng(k):
    ran = random.getrandbits(k)
    NNN.append(ran)
    return ran


def ADD(num):
    NUM = 0
    for i in range(num):
        NUM += Rng(32)
    return NUM

def secure_choice(sequence):
    if not sequence:
        return None
    randbelow = 0
    for i, x in enumerate(sequence):
        randbelow += 1
        if os.urandom(1)[0] < (1 << 8) * randbelow // len(sequence):
            return x

def xDBLADD(P, Q, PQ, q, a, b):
    (X1, Z1), (X2, Z2), (X3, Z3) = PQ, P, Q
    X4 = (X2**2 - a * Z2**2) ** 2 - 8 * b * X2 * Z2**3
    Z4 = 4 * (X2 * Z2 * (X2**2 + a * Z2**2) + b * Z2**4)
    X5 = Z1 * ((X2 * X3 - a * Z2 * Z3) ** 2 - 4 * b * Z2 * Z3 * (X2 * Z3 + X3 * Z2))
    Z5 = X1 * (X2 * Z3 - X3 * Z2) ** 2
    X4, Z4, X5, Z5 = (c % q for c in (X4, Z4, X5, Z5))
    return (X4, Z4), (X5, Z5)

def xMUL(P, k, q, a, b):
    Q, R = (1, 0), P
    for i in reversed(range(k.bit_length() + 1)):
        if k >> i & 1:
            R, Q = Q, R
        Q, R = xDBLADD(Q, R, P, q, a, b)
        if k >> i & 1:
            R, Q = Q, R
    return Q

def shout(x, d, q, a, b):
    P = (x,1)
    Q = xMUL(P, d, q, a, b)
    return Q[0] * pow(Q[1], -1, q) % q

def generate_random_string(length):
    characters = string.ascii_letters + string.digits
    random_string = ''.join(secure_choice(characters) for i in range(length))
    return random_string

class ECCDu():
    def __init__(self,curve,G):
        self.state = getRandomNBitInteger(512)
        self.Curve = curve
        self.g = G

        self.num = Rng(32)
        d = 65537
        self.Q = self.num*self.g
        self.P = d * self.Q

    def ingetkey(self):
        t = int((self.state * self.Q).x())
        self.updade()
        return t%(2**250)

    def updade(self):
        self.state = int((self.state * self.P).x())

    def Random_key(self, n:int):
        out = 0
        number = ceil(n/250)
        for i in range(number):
            out = (out<<250) + self.ingetkey()
        return out % (2**n)

def proof_of_work():
    import tqdm,itertools
    random.seed(os.urandom(8))
    proof = ''.join([random.choice(string.ascii_letters + string.digits) for _ in range(20)])
    _hexdigest = sha256(proof.encode()).hexdigest()
    pr(f"sha256(XXXX+{proof[4:]}) == {_hexdigest}".encode()) 
    pr('Give me XXXX: ')
    for prefix in tqdm.tqdm(itertools.product(string.ascii_letters + string.digits, repeat=4)):
        prefix = ''.join(prefix).encode()
        if hashlib.sha256(prefix+proof[4:].encode()).hexdigest() == _hexdigest:
            x=prefix
            break
    if len(x) != 4 or sha256(x + proof[4:].encode()).hexdigest() != _hexdigest:
        return False
    return True
def get_new_state1(randnums):
    def inverse_right_mask(res, shift, mask=0xffffffff, bits=32):
        tmp = res
        for i in range(bits // shift):
            tmp = res ^ tmp >> shift & mask
        return tmp
    def inverse_left_mask(res, shift, mask=0xffffffff, bits=32):
        tmp = res
        for i in range(bits // shift):
            tmp = res ^ tmp << shift & mask
        return tmp
    def extract_number(y):
        y = y ^ y >> 11
        y = y ^ y << 7 & 2636928640
        y = y ^ y << 15 & 4022730752
        y = y ^ y >> 18
        return y&0xffffffff
    def recover(y):
        y = inverse_right_mask(y,18)
        y = inverse_left_mask(y,15,4022730752)
        y = inverse_left_mask(y,7,2636928640)
        y = inverse_right_mask(y,11)
        return y&0xffffffff
    states = list(map(recover, randnums))
    states2 = []
    for index in range(50):
        i = 32 + index - 20
        y = (states[i] & 0x80000000) + (states[(i + 1) % 624] & 0x7fffffff)
        z = (y >> 1) ^ states[(i + 397) % 624]
        if y % 2 != 0:
            z = z ^ 0x9908b0df
        states2.append(z)
 
    randnums2 = list(map(extract_number, states2))
    new_state1 = sum(randnums2)
    print(f'{new_state1 = }')  
def main():
    pr(BANNER)
    pr('WELCOME TO THIS SIMPLE GAME!!!')
    ASSERT =1 
    #proof_of_work()
    if not ASSERT:
        die("Not right proof of work")

    pr('Now we will start our formal GAME!!!')
    pr('===== First 1💪: =====')
    pr('Enter an integer as the parameter p for Curve: y^2 = x^3+12x+17 (mod p) and 250<p.bit_length()')
    p1 = int(next_prime(2**249))
    if not 250<=p1.bit_length():
        die('Wrong length!')
    E = EllipticCurve(Zmod(p1), [12, 17])
    n = E.order()
    n1 = n // prod([2, 19, 197, 769, 4787, 9341])
    while True:
        G = E.random_point()
        if G.order() == n:
            break
    G = n1 * G
    curve = CurveFp(p1, 12, 17,1)
    pr(curve)
    pr('Please Enter a random_point G:')
    G_t = G
    Gx,Gy = int(G_t[0]),int(G_t[1])
    if not curve.contains_point(Gx,Gy):
        die('This point is outside the curve')
    G = Point(curve,Gx,Gy)
    randnums=[]
    '''
    for i in range(500):
        ECDU = ECCDu(curve,G)
        m  = 'My secret is a random saying of phrase,As below :' + generate_random_string(119)
        Number = ECDU.Random_key(1344)
        c = Number^bytes_to_long(m.encode())
        pr(f'c = {c}')
        pr(f'P = {int(ECDU.P.x()), int(ECDU.P.y())}')
        pr(f'Q = {int(ECDU.Q.x()), int(ECDU.Q.y())}')
        d = 65537
        known = b'My secret is a random saying of phrase,As below :' + b'\x00'*119
        m1 = bytes_to_long(known) ^ c
        prefix = bytes_to_long(long_to_bytes(m1, len(known))[:len(known)-119])<<(119*8)
        number = ceil(1344/250)
        outs = [(prefix>>(i*250)) % 2**250 for i in range(number)][::-1]
        t1=outs[1]
        hs2=outs[2]
        sQ=E.lift_x(ZZ(t1))
        sP=d*sQ
        state=ZZ(sP.xy()[0])
        outs = outs[:2]
        P=str(ECDU.P)
        P=E(eval(P))
        #print(type(P))
        Q=str(ECDU.Q)
        Q=E(eval(Q))
        for i in range(number-2):
            t = int((state*Q).xy()[0])
            state = int((state*P).xy()[0])
            outs.append(t % 2**250)
        out = 0
        for t in outs:
            out = (out<<250) + (t)
        m1 = long_to_bytes(out ^ c)
        pr('Enter m:')
        m_en = m1
        if m_en != m.encode():
            die('This is not the right m,Please try again')
        else:
            pr('Right m!!!')
        G=str(ECDU.g)
        G=E(eval(G))    
        r = G.discrete_log(Q)
        randnums.append(r)
    '''
    for i in range(500):
        randnums.append(Rng(32))
    pr('Bingo!')
    print(len(randnums))
    new_state,new_state1 = ADD(136),ADD(50)
    
    random.seed(new_state)
    _new_state1=get_new_state1(randnums)
    pr('===== Second 2💪: =====')
    curve1 = CurveFp(q,a,b,1)
    pr(f'{int(curve1.p()),int(curve1.a()),int(curve1.b())}')

    pr("Enter a number that does not exceed 1500")
    number = 1700

    pr(f'You only have {number} chances to try')
    success = 0
    leak=[]
    for i in range(1248):
        Gx = secure_choice(select)
        R = Rng(25)
        pr('Gx = ',Gx)
        pr('Px = ',shout(Gx, R, q, a, b))
        R_n = 1
        if R_n != R:
            pr(f'Wrong number!!!,Here is your right number {R}')
            leak.append(R)
        else:
            pr('GGood!')
            success += 1
    import sys
    sys.path.append('./MT19937-Symbolic-Execution-and-Solvermaster/source')
    from MT19937 import MT19937, MT19937_symbolic
    rng_clone = MT19937(state_from_data = (leak, 24))
    outputs = [rng_clone() for _ in range(624)]
    for _ in range(1248 - 624):
        rng_clone()
    for i in range(1700-1248):
        Gx = secure_choice(select)
        R = Rng(25)
        pr('Gx = ',Gx)
        pr('Px = ',shout(Gx, R, q, a, b))
        R_n = rng_clone() >> (32-25)
        if R_n != R:
            pr(f'Wrong number!!!,Here is your right number {R}')
        else:
            pr('GGood!')
            success += 1 
    if int(success)/int(number) <= 0.262:
        die('Please Try more...')
    from z3mt import state2seed624
    seed = state2seed624(outputs)
    _new_state = seed
    random.seed(_new_state)
    iv_n = 0
    for _ in range(2000):
        iv_n += random.getrandbits(32)

    random.seed(new_state)
    iv_num = ADD(2000)

    iv = hashlib.sha256(str(iv_num).encode()).digest()[:16]
    key = hashlib.sha256(str(new_state1).encode()).digest()[:16]
    Aes = AES.new(key,AES.MODE_CBC, iv)
    C = Aes.encrypt(pad(flag,16))

    key_n, iv_n = _new_state1
    iv1 = hashlib.sha256(str(iv_n).encode()).digest()[:16]
    key1 = hashlib.sha256(str(key_n).encode()).digest()[:16]
    Aes_verify = AES.new(key1,AES.MODE_CBC, iv1)
    C_verify = unpad(Aes_verify.decrypt(C),16)

    if C_verify == flag:
        pr('Congratulations🎇🎇🎇!!!!')
        pr('Here is your flag😊：', M)
    else:
        pr('Maybe you could get the right flag!')

if __name__ == '__main__':
    main()

6.electronic_game

题目:

from sage.all import *
from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
from Crypto.Util.number import * 
from base64 import b64encode
from random import * 
from secret import flag
import signal

def _handle_timeout(signum, frame):
    raise TimeoutError('function timeout')

timeout = 66
signal.signal(signal.SIGALRM, _handle_timeout)
signal.alarm(timeout)

def qary_trans_to_int(x, q):
    return sum([int(x[i]) * q**i for i in range(len(x))])

def encode(f, q):
    try:
        return b64encode(long_to_bytes(qary_trans_to_int(f.polynomial().coefficients(sparse = False), q)))
    except:
        return b64encode(long_to_bytes(qary_trans_to_int(f.coefficients(sparse = False), q)))

def generate_irreducible_polynomial(R, n):
    while True: 
        f = R.random_element(degree=n) 
        while f.degree() != n:
            f = R.random_element(degree=n) 
        if f.is_irreducible():
            return f

def generate_sparse_irreducible_polynomial(R, n): 
    x = R.gen()
    while True:
        g = sum(choice([-1, 0, 1]) * x**i for i in range(randint(1, n//2 + 1)))
        if (x**n + g + 1).is_irreducible():
            return x**n + g + 1

def random_polynomial(R, n, beta):  
    return sum(randrange(-beta, beta) * R.gen()**i for i in range(randint(0, n))) + R.gen()**n  

q = 333337
n = 128
beta = 333 
chance = 111
polyns = beta//chance
bound  = 106
R = PolynomialRing(GF(q),'x')

F = generate_irreducible_polynomial(R,n).monic()

k1 = GF(q**n, name = 'a', modulus = generate_sparse_irreducible_polynomial(R,n).monic())
k2 = GF(q**n, name = 'b', modulus = F)

phi = FiniteFieldHomomorphism_generic(Hom(k1, k2)) 

print("F:", encode(F,q).decode())

win_count = 0
for _ in range(chance):
    opt = randint(0, 1)
    if opt:
        As = [phi(random_polynomial(k1,n,beta)) for i in range(polyns)] 
    else:
        As = [k2.random_element() for i in range(polyns)]
    
    for i in range(polyns):
        print(f"As[{i}]: {encode(As[i],q).decode()}")

    opt_guess = input("Guess the option[0/1]: ")
    if int(opt_guess) != opt:
        print("Wrong guess!")
    else:
        win_count += 1
        print("Correct guess!")

if win_count >= bound:
    print("You are so smart! Here is your flag:")
    print(flag)
else:
    print("No flag for you!")

做法:

该题目生成了两个多项式 $F,G$,并在两个多项式以及基域$GF(q)$的基础上生成了两个扩域
其中

$$k_1=GF(q^{128},G)$$ $$k_2=GF(q^{128},F)$$

然后建立了$k_1$到$k_2$的一个同态映射$phi()$
设$A_1$是随机生成的系数较小的多项式(绝对值小于333),$A_2$是扩域$k_2$上的随机多项式

$$opt=1:phi(A_1)$$ $$opt=0:A_2$$

我们需要去根据结果猜opt
看了学长的wp,发现他们使用域中元素的迹来判断opt
域中元素的迹的定义是
$F=\mathbb{F}_{q^m}$是$K=\mathbb{F}_q$的有限扩张
那么，我们有

$$\alpha \in F=\mathbb{F}_{q^m} , K=\mathbb{F}_q$$

$K$上的迹$\bf{Tr}_{F/K}(\alpha)=\alpha + \alpha^{q} + \cdots +\alpha^{q^{m-1}}$
而且有定理可知：

$$\bf{Tr}_{F/K}(\alpha)=-a_{m-1}$$

由于$A_1$的迹很小，且经过同态映射后,迹不发生改变(抽代没学好，不知道为啥)
因此可以通过比较迹来判断 opt 的值
但是我最后复现感觉概率也不是很高，在思考有没有提高概率的方法
复现exp:

from sage.all import *
from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
from Crypto.Util.number import * 
from base64 import *
from random import * 
import signal
flag=b"flag{Great!!!You_have_solved_the_electronic_game!!Keep_going!!}"
def _handle_timeout(signum, frame):
    raise TimeoutError('function timeout')

timeout = 66
signal.signal(signal.SIGALRM, _handle_timeout)
signal.alarm(timeout)

def qary_trans_to_int(x, q):
    return sum([int(x[i]) * q**i for i in range(len(x))])

def encode(f, q):
    try:
        return b64encode(long_to_bytes(qary_trans_to_int(f.polynomial().coefficients(sparse = False), q)))
    except:
        return b64encode(long_to_bytes(qary_trans_to_int(f.coefficients(sparse = False), q)))

def generate_irreducible_polynomial(R, n):
    while True: 
        f = R.random_element(degree=n) 
        while f.degree() != n:
            f = R.random_element(degree=n) 
        if f.is_irreducible():
            return f

def generate_sparse_irreducible_polynomial(R, n): 
    x = R.gen()
    while True:
        g = sum(choice([-1, 0, 1]) * x**i for i in range(randint(1, n//2 + 1)))
        if (x**n + g + 1).is_irreducible():
            return x**n + g + 1

def random_polynomial(R, n, beta):  
    return sum(randrange(-beta, beta) * R.gen()**i for i in range(randint(0, n))) + R.gen()**n  
def decode(encoded_str, q, R):
    decoded_bytes = b64decode(encoded_str)
    integer = bytes_to_long(decoded_bytes)
    coefficients = []
    while integer > 0:
        coefficients.append(integer % q)
        integer //= q
    return R(coefficients)
q=333337
n = 128
beta = 333 
chance = 111
polyns = beta//chance
bound  = 106
R = PolynomialRing(GF(q),'x')

F = generate_irreducible_polynomial(R,n).monic()

k1 = GF(q**n, name = 'a', modulus = generate_sparse_irreducible_polynomial(R,n).monic())
k2 = GF(q**n, name = 'b', modulus = F)

phi = FiniteFieldHomomorphism_generic(Hom(k1, k2)) 

print("F:", encode(F,q).decode())

win_count = 0
for _ in range(chance):
    opt = randint(0, 1)
    if opt:
        As = [phi(random_polynomial(k1,n,beta)) for i in range(polyns)] 
    else:
        As = [k2.random_element() for i in range(polyns)]
    '''
    for i in range(polyns):
        print(f"As[{i}]: {encode(As[i],q).decode()}")
    '''
    GFy = GF(q**n, "y", modulus=F)
    A=[]
    for i in range(polyns):
        i=decode(encode(As[i],q),q,R)
        A.append(GFy(i)) 
    samples = A
    queries = []
    queries1=[]
    for sample in samples:
        trace = int(sample.trace() % q)
        #print(trace)
        if int(trace) - q // 2 < 0:
            # print(trace)
            queries.append(trace<=q//4)
            queries1.append(trace)
        else:
            # print(q - trace)
            queries.append(q - trace<=q//4)
            queries1.append(q-trace)
    '''        
    if any(_>125000 for _ in queries1):
        result = False
    cnt = sum(_>100000 for _ in queries1)
    if cnt == 1 and sum(queries1) < 170000:
        result = True
    '''
    result = sum(queries) > 2
    opt_guess =1 if result else 0 
    if int(opt_guess) != opt:
        print("Wrong guess!")
    else:
        win_count += 1
        print("Correct guess!")

if win_count >= bound:
    print("You are so smart! Here is your flag:")
    print(flag)
else:
    print("No flag for you!")
print(win_count)

7.homomor_game

题目:

import tenseal.sealapi as sealapi
from random import randint
from signal import alarm
import base64

g = 5
poly_modulus_degree = 8192
plain_modulus = 163841

def gen_keys():
	parms = sealapi.EncryptionParameters(sealapi.SCHEME_TYPE.BFV)
	parms.set_poly_modulus_degree(poly_modulus_degree)
	parms.set_plain_modulus(plain_modulus)
	coeff = sealapi.CoeffModulus.BFVDefault(poly_modulus_degree, sealapi.SEC_LEVEL_TYPE.TC128)
	parms.set_coeff_modulus(coeff)
	
	ctx = sealapi.SEALContext(parms, True, sealapi.SEC_LEVEL_TYPE.TC128)
	
	keygen = sealapi.KeyGenerator(ctx)
	public_key = sealapi.PublicKey()
	keygen.create_public_key(public_key)
	secret_key = keygen.secret_key()
	
	parms.save("app/parms")
	public_key.save("app/public_key")
	secret_key.save("app/secret_key")

def load():
	parms = sealapi.EncryptionParameters(sealapi.SCHEME_TYPE.BFV)
	parms.load("app/parms")

	ctx = sealapi.SEALContext(parms, True, sealapi.SEC_LEVEL_TYPE.TC128)

	public_key = sealapi.PublicKey()
	public_key.load(ctx, "app/public_key")

	secret_key = sealapi.SecretKey()
	secret_key.load(ctx, "app/secret_key")
	return ctx, public_key, secret_key

def gen_galois_keys(ctx, secret_key, elt):
	keygen = sealapi.KeyGenerator(ctx, secret_key)
	galois_keys = sealapi.GaloisKeys()
	keygen.create_galois_keys(elt, galois_keys)
	galois_keys.save("app/galois_key")
	return galois_keys

def gen_polynomial(a):
	return '1x^' + str(a)

def f(a, b):
    return (pow(g, a-b, plain_modulus) + pow(g, b-a, plain_modulus) + pow(g, 3*a-b, plain_modulus) + pow(g, a-3*b, plain_modulus))%plain_modulus

def check_result(ctx, decryptor, a, b, pos):
	plaintext = sealapi.Plaintext()
	ciphertext = sealapi.Ciphertext(ctx)
	ciphertext.load(ctx, "app/computation")
	decryptor.decrypt(ciphertext, plaintext)
	assert plaintext[pos] == f(a, b)

def send(filepath):
	f = open(filepath, "rb")
	data = base64.b64encode(f.read()).decode()
	f.close()
	print(data)

def recv(filepath):
	try:
		data = base64.b64decode(input())
	except:
		print("Invalid Base64!")
		exit(0)

	f = open(filepath, "wb")
	f.write(data)
	f.close()

def main():
	ctx, public_key, secret_key = load()
	encryptor = sealapi.Encryptor(ctx, public_key)
	decryptor = sealapi.Decryptor(ctx, secret_key)
	
	a, b = [randint(1, poly_modulus_degree - 1) for _ in range(2)]
	poly_a, poly_b = gen_polynomial(a), gen_polynomial(b)

	plaintext_a, plaintext_b = sealapi.Plaintext(poly_a), sealapi.Plaintext(poly_b)
	ciphertext_a, ciphertext_b = [sealapi.Ciphertext() for _ in range(2)]
	encryptor.encrypt(plaintext_a, ciphertext_a)
	encryptor.encrypt(plaintext_b, ciphertext_b)
	ciphertext_a.save("app/ciphertext_a")
	ciphertext_b.save("app/ciphertext_b")
	galois_used = 0

	alarm(300)

	while True:
		try:
			choice = int(input("Give Me Your Choice:"))
		except:
			print("Invalid choice!")
			exit(0)

		if choice == 0:
			print("Here Is Ciphertext_a:")
			send("app/ciphertext_a")
			print("Here Is Ciphertext_b:")
			send("app/ciphertext_b")

		elif choice == 1:
			if galois_used == 1:
				print("No More!")
				exit(0)
			try:
				elt = int(input("Please give me your choice:"))
			except:
				print("Invalid input!")
				exit(0)

			try:
				galois_key = gen_galois_keys(ctx, secret_key, [elt])
			except:
				print("Invalid galois!")
				exit(0)

			print("Here is your galois key:")
			send("app/galois_key")
			galois_used = 1

		elif choice == 2:
			try:
				pos = int(input("Give Me Your Position:"))
				assert pos in range(poly_modulus_degree)
			except:
				print("Invalid position!")
				exit(0)

			print("Give Me Your Computation")
			recv("app/computation")
	
			try:
				check_result(ctx, decryptor, a, b, pos)
			except:
				print("Incorret Answer!")
				exit(0)

			flag = open("app/flag", "rb")
			print(flag.read())
			flag.close()

		else:
			print("Invalid choice!")
			break

# gen_keys()
main()

做法:

该题生成了一个 BFV 同态加密的例子
choice=0 会给我们单项式$X^{a}$和$X^{b}$的密文 ciphertext_a 和 ciphertext_b
choice=1 会给我们生成一次 galois_key(只有一次)
elt 为 p 的 galois_key 可以建立映射$X^{a} \to X^{ap}$
题目是想让我们凑一个多项式的系数，使得这个系数满足

$$pos=g^{a-b}+g^{b-a}+g^{3a-b}+g^{3b-a} \pmod t$$

所以我们的galois_key的elt可以选择为度数-1,来获得 -a 和 -b
可以先对密文进行运算
得到 $X^{a-b}+X^{b-a}+X^{3a-b}+X^{3b-a}$的密文
然后凑另外一个多项式

$$test(X)=\sum_{i=0}^{N-1}-5^{i}X^{-i}$$

两个多项式进行同态乘之后,常数项即为 pos
(原理,第二个多项式的X正好把原式的X消掉,然后同时多出来与原式X的次数相同的5次方项)
复现exp:

import tenseal.sealapi as sealapi
from random import randint
from signal import alarm
import base64
import sage.all as sa
flag=b"W0w!!!You_actually_solved_the_BFV_Homorize!!!"
g = 5
poly_modulus_degree = 8192
plain_modulus = 163841

def gen_keys():
	parms = sealapi.EncryptionParameters(sealapi.SCHEME_TYPE.BFV)
	parms.set_poly_modulus_degree(poly_modulus_degree)
	parms.set_plain_modulus(plain_modulus)
	coeff = sealapi.CoeffModulus.BFVDefault(poly_modulus_degree, sealapi.SEC_LEVEL_TYPE.TC128)
	parms.set_coeff_modulus(coeff)
	
	ctx = sealapi.SEALContext(parms, True, sealapi.SEC_LEVEL_TYPE.TC128)
	
	keygen = sealapi.KeyGenerator(ctx)
	public_key = sealapi.PublicKey()
	keygen.create_public_key(public_key)
	secret_key = keygen.secret_key()
	
	parms.save("./parms")
	public_key.save("./public_key")
	secret_key.save("./secret_key")

def load():
	parms = sealapi.EncryptionParameters(sealapi.SCHEME_TYPE.BFV)
	parms.load("./parms")

	ctx = sealapi.SEALContext(parms, True, sealapi.SEC_LEVEL_TYPE.TC128)

	public_key = sealapi.PublicKey()
	public_key.load(ctx, "./public_key")

	secret_key = sealapi.SecretKey()
	secret_key.load(ctx, "./secret_key")
	return ctx, public_key, secret_key

def gen_galois_keys(ctx, secret_key, elt):
	keygen = sealapi.KeyGenerator(ctx, secret_key)
	galois_keys = sealapi.GaloisKeys()
	keygen.create_galois_keys(elt, galois_keys)
	galois_keys.save("./galois_key")
	return galois_keys

def gen_polynomial(a):
	return '1x^' + str(a)

def f(a, b):
    return (pow(g, a-b, plain_modulus) + pow(g, b-a, plain_modulus) + pow(g, 3*a-b, plain_modulus) + pow(g, a-3*b, plain_modulus))%plain_modulus

def check_result(ctx, decryptor, a, b, pos):
	plaintext = sealapi.Plaintext()
	ciphertext = sealapi.Ciphertext(ctx)
	ciphertext.load(ctx, "./computation")
	decryptor.decrypt(ciphertext, plaintext)
	assert plaintext[pos] == f(a, b)

def send(filepath):
	f = open(filepath, "rb")
	data = base64.b64encode(f.read()).decode()
	f.close()
	print(data)

def recv(filepath):
	try:
		data = base64.b64decode(input())
	except:
		print("Invalid Base64!")
		exit(0)

	f = open(filepath, "wb")
	f.write(data)
	f.close()
def choose(choice,ctx,secret_key,decryptor,a,b):
    try:
        choice<4 
    except:
        print("Invalid choice!")
        exit(0)

    if choice == 0:
        print("Here Is Ciphertext_a:")
        send("./ciphertext_a")
        print("Here Is Ciphertext_b:")			
        send("./ciphertext_b")

    elif choice == 1:
        galois_used=0
        if galois_used == 1:
            print("No More!")
            exit(0)
        try:
            elt = 8191
        except:
            print("Invalid input!")
            exit(0)

        try:
            galois_key = gen_galois_keys(ctx, secret_key, [elt])
        except:
            print("Invalid galois!")
            exit(0)

        print("Here is your galois key:")
        send("./galois_key")
        galois_used = 1

    elif choice == 2:
        try:
            pos = 0
            assert pos in range(poly_modulus_degree)
        except:
            print("Invalid position!")
            exit(0)

        print("Give Me Your Computation")
        
	
        try:
            check_result(ctx, decryptor, a, b, pos)
        except:
            print("Incorret Answer!")
            exit(0)

        
        print(flag)
        

    else:
        print("Invalid choice!")
	

ctx, public_key, secret_key = load()
encryptor = sealapi.Encryptor(ctx, public_key)
decryptor = sealapi.Decryptor(ctx, secret_key)
a, b = [randint(1, poly_modulus_degree - 1) for _ in range(2)]
poly_a, poly_b = gen_polynomial(a), gen_polynomial(b)

plaintext_a, plaintext_b = sealapi.Plaintext(poly_a), sealapi.Plaintext(poly_b)
ciphertext_a, ciphertext_b = [sealapi.Ciphertext() for _ in range(2)]
encryptor.encrypt(plaintext_a, ciphertext_a)
encryptor.encrypt(plaintext_b, ciphertext_b)
ciphertext_a.save("./ciphertext_a")
ciphertext_b.save("./ciphertext_b")
choose(0,ctx,secret_key,decryptor,a,b)
choose(1,ctx,secret_key,decryptor,a,b)
PR=sa.PolynomialRing(sa.GF(plain_modulus),'x')
x=PR.gen()
mod=x**poly_modulus_degree+1
pol=0
for i in range(poly_modulus_degree-1):
    pol+=-pow(5,i,plain_modulus)*x**(poly_modulus_degree-i)
pol=pol%mod
testx=str(pol)
print(testx)
galois_keys=sealapi.GaloisKeys()
galois_keys.load(ctx,'./galois_key')
evaluator = sealapi.Evaluator(ctx)
ciphertext_minus_a = sealapi.Ciphertext(ctx)
evaluator.apply_galois(ciphertext_a,8191,galois_keys,ciphertext_minus_a)
ciphertext_minus_b = sealapi.Ciphertext(ctx)
evaluator.apply_galois(ciphertext_b,8191,galois_keys,ciphertext_minus_b)

ciphertext_2a = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_a, ciphertext_a, ciphertext_2a)
ciphertext_3a = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_2a, ciphertext_a, ciphertext_3a)

ciphertext_2_minus_b = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_minus_b, ciphertext_minus_b, ciphertext_2_minus_b)
ciphertext_3_minus_b = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_2_minus_b, ciphertext_minus_b, ciphertext_3_minus_b)

ciphertext_3a_minus_b = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_3a, ciphertext_minus_b, ciphertext_3a_minus_b)

ciphertext_a_minus_3b = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_a, ciphertext_3_minus_b, ciphertext_a_minus_3b)

ciphertext_a_minus_b = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_a, ciphertext_minus_b, ciphertext_a_minus_b)

ciphertext_b_minus_a = sealapi.Ciphertext(ctx)
evaluator.multiply(ciphertext_b, ciphertext_minus_a, ciphertext_b_minus_a)
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testx=testx.replace("+"," + ")

plaintext_test=sealapi.Plaintext(testx)

ciphertext_fin1=sealapi.Ciphertext()
evaluator.add(ciphertext_a_minus_b,ciphertext_b_minus_a,ciphertext_fin1)

ciphertext_fin2=sealapi.Ciphertext()
evaluator.add(ciphertext_3a_minus_b,ciphertext_a_minus_3b,ciphertext_fin2)

ciphertext_fin3=sealapi.Ciphertext()
evaluator.add(ciphertext_fin1,ciphertext_fin2,ciphertext_fin3)

ciphertext_fin4=sealapi.Ciphertext()
evaluator.multiply_plain(ciphertext_fin3,plaintext_test,ciphertext_fin4)
ciphertext_fin4.save("./computation")
choose(2,ctx,secret_key,decryptor,a,b)

结语

这是第一次完整的复现难度高的比赛题了，中间写写停停，不过好在没有放弃，用了一周的时间完成复现，还需继续加油