第四届“网鼎杯”网络安全大赛 - 青龙组

Crypto

CRYPTO01

2023 年江苏省领航杯 bd 原题：

题目：

from Crypto.Util.number import *
from secret import flag

p = getPrime(512)
q = getPrime(512)
n = p * q
d = getPrime(299)
e = inverse(d,(p-1)*(q-1))
m = bytes_to_long(flag)
c = pow(m,e,n)
hint1 = p >> (512-70)
hint2 = q >> (512-70)

print(f"n = {
     n}")
print(f"e = {
     e}")
print(f"c = {
     c}")
print(f"hint1 = {
     hint1}")
print(f"hint2 = {
     hint2}")

n = 95987463597889741532025162535631829592517704738860431905943824498597890101136796870879646153634795544527837591685182170270252555997933421564167468816667980089869165228796395618775798781717091178143300536302805947806332962230499807469654672313206953750808878098101882367253566423367338396717229488061237787619
e = 12761568528114005244342182138319275328501544878744699681091257281459893043102455333575012392492554249378138377894473691530001400901111860569849400611145049669197371290112261284014768486937294128410391332641648640840309135718123516725634190495261570498188176102281061084366456374907755950527213961595506707585
c = 35154471719082941146017277238175991504655570882040897713927696748547265178059291385527810510087300325678561502592029686486144851647754163110759942860051150957340793828236482243293747881251009434747066136505663782280758979333019882382608002308773317613847702048959887017010151916544703971113441862351668075919
hint1 = 1175980694459189065778
hint2 = 632846170973644915854

思路：

利用上高位的boneh_durfee攻击

参考论文：

[367.pdf (iacr.org)](https://eprint.iacr.org/2023/367.pdf)

他们实验时的脚本：Codes of the manuscript—Practical Attacks on Small Private Exponent RSA—New Records and New Insi - Pastebin.com

exp:

先恢复 d

import time
time.clock = time.time
 
debug = True
 
strict = False
 
helpful_only = True
dimension_min = 7 # 如果晶格达到该尺寸，则停止移除
# 显示有用矢量的统计数据
def helpful_vectors(BB, modulus):
    nothelpful = 0
    for ii in range(BB.dimensions()[0]):
        if BB[ii,ii] >= modulus:
            nothelpful += 1

# 显示带有 0 和 X 的矩阵
def matrix_overview(BB, bound):
    for ii in range(BB.dimensions()[0]):
        a = ('%02d ' % ii)
        for jj in range(BB.dimensions()[1]):
            a += '0' if BB[ii,jj] == 0 else 'X'
            if BB.dimensions()[0] < 60: 
                a += ' '
        if BB[ii, ii] >= bound:
            a += '~'
        #print (a)

# 尝试删除无用的向量
# 从当前 = n-1（最后一个向量）开始
def remove_unhelpful(BB, monomials, bound, current):
    # 我们从当前 = n-1（最后一个向量）开始
    if current == -1 or BB.dimensions()[0] <= dimension_min:
        return BB
 
    # 开始从后面检查
    for ii in range(current, -1, -1):
        #  如果它没有用
        if BB[ii, ii] >= bound:
            affected_vectors = 0
            affected_vector_index = 0
             # 让我们检查它是否影响其他向量
            for jj in range(ii + 1, BB.dimensions()[0]):
                # 如果另一个向量受到影响：
                # 我们增加计数
                if BB[jj, ii] != 0:
                    affected_vectors += 1
                    affected_vector_index = jj
 
            # 等级：0
            # 如果没有其他载体最终受到影响
            # 我们删除它
            if affected_vectors == 0:
                #print ("* removing unhelpful vector", ii)
                BB = BB.delete_columns([ii])
                BB = BB.delete_rows([ii])
                monomials.pop(ii)
                BB = remove_unhelpful(BB, monomials, bound, ii-1)
                return BB
 
           # 等级：1
            #如果只有一个受到影响，我们会检查
            # 如果它正在影响别的向量
            elif affected_vectors == 1:
                affected_deeper = True
                for kk in range(affected_vector_index + 1, BB.dimensions()[0]):
                    # 如果它影响哪怕一个向量
                    # 我们放弃这个
                    if BB[kk, affected_vector_index] != 0:
                        affected_deeper = False
                # 如果没有其他向量受到影响，则将其删除，并且
                # 这个有用的向量不够有用
                #与我们无用的相比
                if affected_deeper and abs(bound - BB[affected_vector_index, affected_vector_index]) < abs(bound - BB[ii, ii]):
                    #print ("* removing unhelpful vectors", ii, "and", affected_vector_index)
                    BB = BB.delete_columns([affected_vector_index, ii])
                    BB = BB.delete_rows([affected_vector_index, ii])
                    monomials.pop(affected_vector_index)
                    monomials.pop(ii)
                    BB = remove_unhelpful(BB, monomials, bound, ii-1)
                    return BB
    # nothing happened
    return BB
 
""" 
Returns:
* 0,0   if it fails
* -1，-1 如果 "strict=true"，并且行列式不受约束
* x0,y0 the solutions of `pol`
"""
def boneh_durfee(pol, modulus, mm, tt, XX, YY):
    """
    Boneh and Durfee revisited by Herrmann and May
 
 在以下情况下找到解决方案：
* d < N^delta
* |x|< e^delta
* |y|< e^0.5
每当 delta < 1 - sqrt（2）/2 ~ 0.292
    """
 
    # substitution (Herrman and May)
    PR.<u, x, y> = PolynomialRing(ZZ)   #多项式环
    Q = PR.quotient(x*y + 1 - u)        #  u = xy + 1
    polZ = Q(pol).lift()
 
    UU = XX*YY + 1
 
    # x-移位
    gg = []
    for kk in range(mm + 1):
        for ii in range(mm - kk + 1):
            xshift = x^ii * modulus^(mm - kk) * polZ(u, x, y)^kk
            gg.append(xshift)
    gg.sort()
 
    # 单项式 x 移位列表
    monomials = []
    for polynomial in gg:
        for monomial in polynomial.monomials(): #对于多项式中的单项式。单项式（）：
            if monomial not in monomials:  # 如果单项不在单项中
                monomials.append(monomial)
    monomials.sort()
 
    # y-移位
    for jj in range(1, tt + 1