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虽然离比赛结束已经隔了很久了，而且当时自己也没有打，但还是想抽时间记录下一下crypto题目，正所谓书山有路勤为径，学海无涯苦作舟。

complex_signin

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from Crypto.Util.number import *
from Crypto.Cipher import ChaCha20
import hashlib
from secret import flag

class Complex:
    def __init__(self, re, im):
        self.re = re
        self.im = im

    def __mul__(self, c):
        re_ = self.re * c.re - self.im * c.im
        im_ = self.re * c.im + self.im * c.re
        return Complex(re_, im_)

    def __eq__(self, c):
        return self.re == c.re and self.im == c.im

    def __rshift__(self, m):
        return Complex(self.re >> m, self.im >> m)

    def __lshift__(self, m):
        return Complex(self.re << m, self.im << m)

    def __str__(self):
        if self.im == 0:
            return str(self.re)
        elif self.re == 0:
            if abs(self.im) == 1:
                return f"{'-' if self.im < 0 else ''}i"
            else:
                return f"{self.im}i"
        else:
            return f"{self.re} {'+' if self.im > 0 else '-'} {abs(self.im)}i"

    def tolist(self):
        return [self.re, self.im]

def complex_pow(c, exp, n):
    result = Complex(1, 0)
    while exp > 0:
        if exp & 1:
            result = result * c
            result.re = result.re % n
            result.im = result.im % n
        c = c * c
        c.re = c.re % n
        c.im = c.im % n
        exp >>= 1
    return result

bits = 128
p = getPrime(1024)
q = getPrime(1024)
n = p * q
m = Complex(getRandomRange(1, n), getRandomRange(1, n))
e = 3
c = complex_pow(m, e, n)
print(f"n = {n}")
print(f"mh = {(m >> bits << bits).tolist()}")
print(f"C = {c.tolist()}")
print(f"enc = {ChaCha20.new(key=hashlib.sha256(str(m.re + m.im).encode()).digest(), nonce=b'Pr3d1ctmyxjj').encrypt(flag)}")

'''
n = 24240993137357567658677097076762157882987659874601064738608971893024559525024581362454897599976003248892339463673241756118600994494150721789525924054960470762499808771760690211841936903839232109208099640507210141111314563007924046946402216384360405445595854947145800754365717704762310092558089455516189533635318084532202438477871458797287721022389909953190113597425964395222426700352859740293834121123138183367554858896124509695602915312917886769066254219381427385100688110915129283949340133524365403188753735534290512113201932620106585043122707355381551006014647469884010069878477179147719913280272028376706421104753
mh = [3960604425233637243960750976884707892473356737965752732899783806146911898367312949419828751012380013933993271701949681295313483782313836179989146607655230162315784541236731368582965456428944524621026385297377746108440938677401125816586119588080150103855075450874206012903009942468340296995700270449643148025957527925452034647677446705198250167222150181312718642480834399766134519333316989347221448685711220842032010517045985044813674426104295710015607450682205211098779229647334749706043180512861889295899050427257721209370423421046811102682648967375219936664246584194224745761842962418864084904820764122207293014016, 15053801146135239412812153100772352976861411085516247673065559201085791622602365389885455357620354025972053252939439247746724492130435830816513505615952791448705492885525709421224584364037704802923497222819113629874137050874966691886390837364018702981146413066712287361010611405028353728676772998972695270707666289161746024725705731676511793934556785324668045957177856807914741189938780850108643929261692799397326838812262009873072175627051209104209229233754715491428364039564130435227582042666464866336424773552304555244949976525797616679252470574006820212465924134763386213550360175810288209936288398862565142167552]
C = [5300743174999795329371527870190100703154639960450575575101738225528814331152637733729613419201898994386548816504858409726318742419169717222702404409496156167283354163362729304279553214510160589336672463972767842604886866159600567533436626931810981418193227593758688610512556391129176234307448758534506432755113432411099690991453452199653214054901093242337700880661006486138424743085527911347931571730473582051987520447237586885119205422668971876488684708196255266536680083835972668749902212285032756286424244284136941767752754078598830317271949981378674176685159516777247305970365843616105513456452993199192823148760, 21112179095014976702043514329117175747825140730885731533311755299178008997398851800028751416090265195760178867626233456642594578588007570838933135396672730765007160135908314028300141127837769297682479678972455077606519053977383739500664851033908924293990399261838079993207621314584108891814038236135637105408310569002463379136544773406496600396931819980400197333039720344346032547489037834427091233045574086625061748398991041014394602237400713218611015436866842699640680804906008370869021545517947588322083793581852529192500912579560094015867120212711242523672548392160514345774299568940390940653232489808850407256752]
enc = b'\x9c\xc4n\x8dF\xd9\x9e\xf4\x05\x82!\xde\xfe\x012$\xd0\x8c\xaf\xfb\rEb(\x04)\xa1\xa6\xbaI2J\xd2\xb2\x898\x11\xe6x\xa9\x19\x00pn\xf6rs- \xd2\xd1\xbe\xc7\xf51.\xd4\xd2 \xe7\xc6\xca\xe5\x19\xbe'
'''

思路分析：涉及二元copper，复数加密

对复数m=a+bi进行rsa加密，这里的e很小，所以直接化简：

我们知道a和b高1024-180位的结果，已知，显然可以使用coppersmith定理恢复出低位信息，但这里有两个未知数a和b，所以使用二元copper（这里只对实部进行copper即可）

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f = (a_high+x)^3-3*(a_high+x)*(b_high+y)^2-c[0]

得到m后，反向解chacha20即可

EXP：

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from Crypto.Util.number import *
from Crypto.Cipher import ChaCha20
import hashlib
import itertools
def small_roots(f, bounds, m=1, d=None):
    if not d:
        d = f.degree()

    R = f.base_ring()
    N = R.cardinality()

    f /= f.coefficients().pop(0)
    f = f.change_ring(ZZ)

    G = Sequence([], f.parent())
    for i in range(m + 1):
        base = N ^ (m - i) * f ^ i
        for shifts in itertools.product(range(d), repeat=f.nvariables()):
            g = base * prod(map(power, f.variables(), shifts))
            G.append(g)

    B, monomials = G.coefficient_matrix()
    monomials = vector(monomials)

    factors = [monomial(*bounds) for monomial in monomials]
    for i, factor in enumerate(factors):
        B.rescale_col(i, factor)

    B = B.dense_matrix().LLL()

    B = B.change_ring(QQ)
    for i, factor in enumerate(factors):
        B.rescale_col(i, 1 / factor)

    H = Sequence([], f.parent().change_ring(QQ))
    for h in filter(None, B * monomials):
        H.append(h)
        I = H.ideal()
        if I.dimension() == -1:
            H.pop()
        elif I.dimension() == 0:
            roots = []
            for root in I.variety(ring=ZZ):
                root = tuple(R(root[var]) for var in f.variables())
                roots.append(root)
            return roots

    return []


a_high = 3960604425233637243960750976884707892473356737965752732899783806146911898367312949419828751012380013933993271701949681295313483782313836179989146607655230162315784541236731368582965456428944524621026385297377746108440938677401125816586119588080150103855075450874206012903009942468340296995700270449643148025957527925452034647677446705198250167222150181312718642480834399766134519333316989347221448685711220842032010517045985044813674426104295710015607450682205211098779229647334749706043180512861889295899050427257721209370423421046811102682648967375219936664246584194224745761842962418864084904820764122207293014016
b_high = 15053801146135239412812153100772352976861411085516247673065559201085791622602365389885455357620354025972053252939439247746724492130435830816513505615952791448705492885525709421224584364037704802923497222819113629874137050874966691886390837364018702981146413066712287361010611405028353728676772998972695270707666289161746024725705731676511793934556785324668045957177856807914741189938780850108643929261692799397326838812262009873072175627051209104209229233754715491428364039564130435227582042666464866336424773552304555244949976525797616679252470574006820212465924134763386213550360175810288209936288398862565142167552
n = 24240993137357567658677097076762157882987659874601064738608971893024559525024581362454897599976003248892339463673241756118600994494150721789525924054960470762499808771760690211841936903839232109208099640507210141111314563007924046946402216384360405445595854947145800754365717704762310092558089455516189533635318084532202438477871458797287721022389909953190113597425964395222426700352859740293834121123138183367554858896124509695602915312917886769066254219381427385100688110915129283949340133524365403188753735534290512113201932620106585043122707355381551006014647469884010069878477179147719913280272028376706421104753
c = [5300743174999795329371527870190100703154639960450575575101738225528814331152637733729613419201898994386548816504858409726318742419169717222702404409496156167283354163362729304279553214510160589336672463972767842604886866159600567533436626931810981418193227593758688610512556391129176234307448758534506432755113432411099690991453452199653214054901093242337700880661006486138424743085527911347931571730473582051987520447237586885119205422668971876488684708196255266536680083835972668749902212285032756286424244284136941767752754078598830317271949981378674176685159516777247305970365843616105513456452993199192823148760, 21112179095014976702043514329117175747825140730885731533311755299178008997398851800028751416090265195760178867626233456642594578588007570838933135396672730765007160135908314028300141127837769297682479678972455077606519053977383739500664851033908924293990399261838079993207621314584108891814038236135637105408310569002463379136544773406496600396931819980400197333039720344346032547489037834427091233045574086625061748398991041014394602237400713218611015436866842699640680804906008370869021545517947588322083793581852529192500912579560094015867120212711242523672548392160514345774299568940390940653232489808850407256752]
P.<x,y>=PolynomialRing(Zmod(n))

f = (a_high+x)^3-3*(a_high+x)*(b_high+y)^2-c[0]
a_low , b_low=small_roots(f,[2**128,2**128],3)[0]

print("a =",a_high+a_low)
print("b =",b_high+b_low)

a=a_high+a_low
b=b_high+b_low

enc = b'\x9c\xc4n\x8dF\xd9\x9e\xf4\x05\x82!\xde\xfe\x012$\xd0\x8c\xaf\xfb\rEb(\x04)\xa1\xa6\xbaI2J\xd2\xb2\x898\x11\xe6x\xa9\x19\x00pn\xf6rs- \xd2\xd1\xbe\xc7\xf51.\xd4\xd2 \xe7\xc6\xca\xe5\x19\xbe'

key = hashlib.sha256(str(a + b).encode()).digest()
nonce = b'Pr3d1ctmyxjj'

cipher = ChaCha20.new(key=key, nonce=nonce)
flag = cipher.decrypt(enc)

print(f"flag = {flag}")

#XYCTF{Welcome_to_XYCTF_Now_let_us_together_play_Crypto_challenge}