Crypto日记之picoCTF 2022中的RSA题目求解_from binascii import hexlify from gmpy2 import * i-优快云博客

本文链接：https://blog.youkuaiyun.com/zgzhzywzd/article/details/123577222

本文详细介绍了在picoCTF 2022竞赛中遇到的三道RSA加密题目，包括Very Smooth、Sum-O-Primes和NSA Backdoor的解题过程。通过分析代码，运用Pollard's p-1算法成功分解了n，并利用给定信息解密。文章提供了具体的Python代码实现，展示了如何解决整数分解和离散对数问题。

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前言

picoCTF 2022一共66道题，队友基本都解出来了，只能说师傅们tql！共有15道Cryptography题目，其中3道RSA题目，难点应该都在整数分解的算法，借Generate师傅的wp分析一波，靠谱！

题解

0x01 Very Smooth

Description：Forget safe primes… Here, we like to live life dangerously… >:)
看下代码：

#!/usr/bin/python

from binascii import hexlify
from gmpy2 import *
import math
import os
import sys

if sys.version_info < (3, 9):
    math.gcd = gcd
    math.lcm = lcm

_DEBUG = False

FLAG  = open('flag.txt').read().strip()
FLAG  = mpz(hexlify(FLAG.encode()), 16)
SEED  = mpz(hexlify(os.urandom(32)).decode(), 16)
STATE = random_state(SEED)

def get_prime(state, bits):
    return next_prime(mpz_urandomb(state, bits) | (1 << (bits - 1)))

def get_smooth_prime(state, bits, smoothness=16):
    p = mpz(2)
    p_factors = [p]
    while p.bit_length() < bits - 2 * smoothness:
        factor = get_prime(state, smoothness)
        p_factors.append(factor)
        p *= factor

    bitcnt = (bits - p.bit_length()) // 2
    while True:
        prime1 = get_prime(state, bitcnt)
        prime2 = get_prime(state, bitcnt)
        tmpp = p * prime1 * prime2
        if tmpp.bit_length() < bits:
            bitcnt += 1
            continue
        if tmpp.bit_length() > bits:
            bitcnt -= 1
            continue
        if is_prime(tmpp + 1):
            p_factors.append(prime1)
            p_factors.append(prime2)
            p = tmpp + 1
            break
    p_factors.sort()
    return (p, p_factors)

e = 0x10001

while True:
    p, p_factors = get_smooth_prime(STATE, 1024, 16)
    if len(p_factors) != len(set(p_factors)):
        continue
    # Smoothness should be different or some might encounter issues.
    q, q_factors = get_smooth_prime(STATE, 1024, 17)
    if len(q_factors) != len(set(q_factors)):
        continue
    factors = p_factors + q_factors
    if e not in factors:
        break

if _DEBUG:
    import sys
    sys.stderr.write(f'p = {
     
     p.digits(16)}\n\n')
    sys.stderr.write(f'p_factors = [\n')
    for factor in p_factors:
        sys.stderr.write(f'    {
     
     factor.digits(16)},\n')
    sys.s