已知n与phi分解n

题目:

from Crypto.Util.number import getPrime
from math import prod
from sympy import nextprime
from random import choices

with open('flag.txt', 'rb') as fs:
    flag = fs.read().strip()

primes = [getPrime(512) for _ in range(9)]
print(f"{prod(primes) = }")
print(f"{prod(p - 1 for p in primes) = }")

primes2 = [nextprime(p) for p in choices(primes, k=3)]
n = prod(primes2)
e = 65537
c = pow(int.from_bytes(flag, 'big'), e, n)

print(f'n = {n}')
print(f'e = {e}')
print(f'c = {c}')

# 363364907814244019888662301376841344262476227242899756862391470731421569394957444030214887114615748277199649349781524749919652160244484352285668794188836866602305788131186220057989320357344904731322223310531945208433910803617954798258382169132907508787682006064930747033681966462568715421005454243255297306718356766130469885581576362173340673516476386201173298433892314145854649884922769732583885904512624543994675379894718657682146178638074984373206937523380103438050549181568015985546172618830480078894445808092527561363650503540062128543705172678754195578429520889784813733491180748361345720247750720179608752244490362713103319685024237941527268458213442611663415417005556439749055222361212059968254748751273361732365487788593341859760309778894350385339764442343374673786357175846291309425081492959910254127778240522152676060766139057453197528944251599979227271074508795482632471242983094008619339488744362509349734218480932255216087706001484182136783834973304870508270118505737767002256270427907341952256516206663258530300791364944105025764611810001781971638030661367630116818647252727909489405550104641122269772492252464714694507693447974171377200402508765841829763548525530878309985480248379655169722567051495205792089930014228403456098065971372039443284193603395249634283366194562380309469628114581468645669390610963076340643757972439104287127375438663839421605531570285615180251
# 363364907814244019888662301376841344262476227242899756862391470731421569394957444030214887114615748277199649349781524749919652160244484352285668794188836492373364350673588273863828369502073826782362255108313852264064760467561392054178047091483873483255491431451728274259516789065331176728192953741805933100379191778599394515981288225535175013258094287912195847642598436035132783919453991516358280321085873745330313812205910011387125778714795906023110368957596998222544234082487264006696812862179916726781327290284827659294751262185328816323311831349296593013038823107653943652771448719760448938995150646738377177532550757319539185878535087009904848382493668686831331474113789651777885239747000076063679062106375348803749466079052774597412239427050432901553466002731972993029311850718200685157193170716432600165476733200831046297530470544781309612128231925681374239849452623513538498417735984094919756374577623486416462101457492789215144166273775249387638107644634704270216130852885082174564648445147377239033930079759024399532146184753110240154062693457622208373371290126810856885343328090305620627668495081760346853701632815149478447405718664667978825807101325764916405446176183238866136433205933785973568759281210319422288153910340542098573782006262190181726245838857185687242960093445000287347616796984610291664809895901301187179157382169999966124177588884152267266994164841066291200
# n = 899081756851564072995842371038848265712822308942406479625157544735473115850983700580364485532298999127834142923262920189902691972009898741820291331257478170998867183390650298055916005944577877856728843264502218692432679062445730259562784479410120575777748292393321588239071577384218317338474855507210816917917699500763270490789679076190405915250953860114858086078092945282693720016414837231157788381144668395364877545151382171251673050910143023561541226464220441
# e = 65537
# c = 841335863342518623856757469220437045493934999201203757845757404101093751603513457430254875658199946020695655428637035628085973393246970440054477600379027466651143466332405520374224855994531411584946074861018245519106776529260649700756908093025092104292223745612991818151040610497258923925952531383407297026038305824754456660932812929344928080812670596607694776017112795053283695891798940700646874515366341575417161087304105309794441077774052357656529143940010140

解题思路:

解答:

from Crypto.Util.number import *
from gmpy2 import *
from sympy import nextprime

N = 363364907814244019888662301376841344262476227242899756862391470731421569394957444030214887114615748277199649349781524749919652160244484352285668794188836866602305788131186220057989320357344904731322223310531945208433910803617954798258382169132907508787682006064930747033681966462568715421005454243255297306718356766130469885581576362173340673516476386201173298433892314145854649884922769732583885904512624543994675379894718657682146178638074984373206937523380103438050549181568015985546172618830480078894445808092527561363650503540062128543705172678754195578429520889784813733491180748361345720247750720179608752244490362713103319685024237941527268458213442611663415417005556439749055222361212059968254748751273361732365487788593341859760309778894350385339764442343374673786357175846291309425081492959910254127778240522152676060766139057453197528944251599979227271074508795482632471242983094008619339488744362509349734218480932255216087706001484182136783834973304870508270118505737767002256270427907341952256516206663258530300791364944105025764611810001781971638030661367630116818647252727909489405550104641122269772492252464714694507693447974171377200402508765841829763548525530878309985480248379655169722567051495205792089930014228403456098065971372039443284193603395249634283366194562380309469628114581468645669390610963076340643757972439104287127375438663839421605531570285615180251
phi = 363364907814244019888662301376841344262476227242899756862391470731421569394957444030214887114615748277199649349781524749919652160244484352285668794188836492373364350673588273863828369502073826782362255108313852264064760467561392054178047091483873483255491431451728274259516789065331176728192953741805933100379191778599394515981288225535175013258094287912195847642598436035132783919453991516358280321085873745330313812205910011387125778714795906023110368957596998222544234082487264006696812862179916726781327290284827659294751262185328816323311831349296593013038823107653943652771448719760448938995150646738377177532550757319539185878535087009904848382493668686831331474113789651777885239747000076063679062106375348803749466079052774597412239427050432901553466002731972993029311850718200685157193170716432600165476733200831046297530470544781309612128231925681374239849452623513538498417735984094919756374577623486416462101457492789215144166273775249387638107644634704270216130852885082174564648445147377239033930079759024399532146184753110240154062693457622208373371290126810856885343328090305620627668495081760346853701632815149478447405718664667978825807101325764916405446176183238866136433205933785973568759281210319422288153910340542098573782006262190181726245838857185687242960093445000287347616796984610291664809895901301187179157382169999966124177588884152267266994164841066291200
n = 899081756851564072995842371038848265712822308942406479625157544735473115850983700580364485532298999127834142923262920189902691972009898741820291331257478170998867183390650298055916005944577877856728843264502218692432679062445730259562784479410120575777748292393321588239071577384218317338474855507210816917917699500763270490789679076190405915250953860114858086078092945282693720016414837231157788381144668395364877545151382171251673050910143023561541226464220441
e = 65537
c = 841335863342518623856757469220437045493934999201203757845757404101093751603513457430254875658199946020695655428637035628085973393246970440054477600379027466651143466332405520374224855994531411584946074861018245519106776529260649700756908093025092104292223745612991818151040610497258923925952531383407297026038305824754456660932812929344928080812670596607694776017112795053283695891798940700646874515366341575417161087304105309794441077774052357656529143940010140

from random import randrange


def factorize(N, phi):
    """
    Recovers the prime factors from a modulus if Euler's totient is known.
    This method only works for a modulus consisting of 2 primes!
    :param N: the modulus
    :param phi: Euler's totient, the order of the multiplicative group modulo N
    :return: a tuple containing the prime factors, or None if the factors were not found
    """
    s = N + 1 - phi
    d = s ** 2 - 4 * N
    p = int(s - iroot(d,2)[0]) // 2
    q = int(s + iroot(d,2)[0]) // 2
    return p, q


def factorize_multi_prime(N, phi):
    """
    Recovers the prime factors from a modulus if Euler's totient is known.
    This method works for a modulus consisting of any number of primes, but is considerably be slower than factorize.
    More information: Hinek M. J., Low M. K., Teske E., "On Some Attacks on Multi-prime RSA" (Section 3)
    :param N: the modulus
    :param phi: Euler's totient, the order of the multiplicative group modulo N
    :return: a tuple containing the prime factors
    """
    prime_factors = set()
    factors = [N]
    while len(factors) > 0:
        # Element to factorize.
        N = factors[0]

        w = randrange(2, N - 1)
        i = 1
        while phi % (2 ** i) == 0:
            sqrt_1 = pow(w, phi // (2 ** i), N)
            if sqrt_1 > 1 and sqrt_1 != N - 1:
                # We can remove the element to factorize now, because we have a factorization.
                factors = factors[1:]

                p = gcd(int(N), int(sqrt_1 + 1))
                q = N // p

                if is_prime(p):
                    prime_factors.add(p)
                elif p > 1:
                    factors.append(p)

                if is_prime(q):
                    prime_factors.add(q)
                elif q > 1:
                    factors.append(q)

                # Continue in the outer loop
                break

            i += 1

    return tuple(prime_factors)

primelist = factorize_multi_prime(N,phi)
plist = []
for i in primelist:
    if(n % nextprime(i) == 0):
        plist.append(nextprime(i))

phi_n = 1
for i in plist:
    phi_n *= (i-1)
d = inverse(e,phi_n)
m = int(pow(c,d,n))
print(long_to_bytes(m))
#moectf{you_KNow_how_to_faCtorize_N_right?_9?WPIBung6?WPIBung6?WPIBund6?}