2020CISCN初赛-crypto-lfsr writeup

第十三届全国大学生信息安全竞赛初赛，密码学的lfsr详解。

原题文件：https://wwe.lanzous.com/iDF95gntw5a

本题考查线性反馈移位寄存器。

初始状态，反馈函数，输出序列都已给出，已知n是100。

输出序列即题给的output.txt。
反馈函数是代码形式，需要经过分析后转化为表达式形式。
通过分析反馈函数可以发现，初始状态即是输出序列的前100位倒置。

lfsr源代码如下，

def lfsr(state, mask):
    feedback = state & mask
    feed_bit = bin(feedback)[2:].count("1") & 1  #当feedback中有奇数个1时，feed_bit为1
    output_bit = state & 1  #取state的最后一位
    state = (state >> 1) | (feed_bit << (N-1))  #state右移一位，将feed_bit反馈到state的第一位
    return state, output_bit

其中feed_bit是反馈位，当feedback中有奇数个1时，feed_bit为1。
因为0异或任何数等于任何数本身，不影响最后结果，则当feedback中有奇数个1时，最低位向最高位依次做异或运算的结果是1。

因此反馈位feed_bit就是将当前状态的每一位与mask中对应位做位与，并将每一位计算结果依次异或。

设lfsr每一位依次为a1到a100，mask中每一位依次为b1到b100，可得反馈函数如下：

$$feed\_bit=b_{100}\&a_{100}\oplus b_{99}\&a_{99}\oplus b_{98}\&a_{98}\oplus b_{97}\&a_{97}\oplus \dots \dots \oplus b_1\&a_1$$

output_bit是输出位，取lfsr当前状态的最后一位输出。

state表示当前状态，每次state右移一位，并将feed_bit反馈到state的第一位。

整个lfsr流程如下图：

因此我们由输出序列可以知道lfsr初始状态，和其后的每一个反馈位。

本题将flag设为mask，即要求mask。

根据反馈函数，可知mask的每一位都是反馈函数的系数。
而反馈函数的值与参数都已知，我们可以通过列100个反馈函数表达式，利用矩阵解线性方程组，计算得到100个系数的值。

由于在有限域GF(2)内，只有0和1，此时乘法相当于异或，加法相当于与运算。
因此可直接将公式中的异或（⊕）替换为乘法（*），将与运算（&）替换为加法（+）。

计算原理如下（c表示输出序列）：

$$\begin{array}{rl}& c_{101}=b_{100}\ast c_{100}+b_{99}\ast c_{99}+b_{98}\ast c_{98}+b_{97}\ast c_{97}+\dots \dots +b_1\ast c_1\\ & c_{102}=b_{100}\ast c_{101}+b_{99}\ast c_{100}+b_{98}\ast c_{99}+b_{97}\ast c_{98}+\dots \dots +b_1\ast c_2\\ & ......\\ & c_{200}=b_{100}\ast c_{199}+b_{99}\ast c_{198}+b_{98}\ast c_{197}+b_{97}\ast c_{196}+\dots \dots +b_1\ast c_{100}\end{array}$$
$$\left[\begin{array}{cccc}{c}_{100}& c_{99}& ...& c_1\\ c_{101}& c_{100}& ...& c_2\\ ...& ...& & ...\\ c_{199}& c_{198}& ...& c_{100}\end{array}\right]\ast \left[\begin{array}{c}{b}_{100}\\ b_{99}\\ ...\\ b_1\end{array}\right]=\left[\begin{array}{c}{c}_{101}\\ c_{102}\\ ...\\ c_{200}\end{array}\right]$$

sage脚本为：

from sage.all_cmdline import *
import hashlib

output = '0100110011101111011111011010100111001010010100001111110110111101011110011111010010010111011100110111011001100001010010011101101000110110000011111011110000010100001001000011001011011011011001111110101111001010011011100010011110101100100101111001011100011110111101001010000000100100000111100101100100100000100110001111110011100110101000110100011001110110000110111111010111000001101110011001101111010110101000101011110111001011001111111000100100100011100100010101001100111011101111100110101011101101001001110010111011010011111001100001111111100001000001101101000001010100110011001101000101001010001001011000010111111100101111000000101000010110100110101001001100100011000111101000110011110101101111100110101010101000000001110010101001001111101001010111011110110001100101001111110111000110101011011010110101100011110010000011000110010111101010010000011101110111100011011110110001111010101011101100101101001111100100011000100101111011110001111100000110111000111111000111100000110011011011111110011111100010010110000010011100001010101011011101100011001100101100011001010010001101010011000000011110110111000000110001010010111110101011100000010100100001101011010101100110000101100010001010010111011000100001000001001110101011000010111000000010101100001010000010110110000111100011011000010111001111101110111010110000011001011101100000100000100010100100000010011010000111110010001111010000100011110000101111000110100010110101110001011101101100111000001110100100000000010011001110101110000001101100000001010111010011111000101101011001010011001101000111000010111111010101000011011001001110000011100100100010110101100001110110111001000001011111100101100011010010010110000011110100110000111011011010101110101001100110100101100101000100011001011000001011110010000011000110111011110011010001101100010010111110101101101110100111010000001101000011011010011110111101000000100011000000101001000000011010101010000100001110110000110011000111101011100100101111101111111110101000011010001101100010010100111000000010000101100100110101000001001011100011111001101001101100010100000111010101111011101111111101100000010000111110011101010100101110100101010100011110100100000111010001000100011010011100010011000001110110100110110010100100110111011001010010010100000110101110000011110010110100010101000110000111101011011010111101100111100101001101001101100111101000100101101100010111010010111100100111101101110101110011010110011111011111010000011000100010010101110001001000111111100100010100000001001010111010100010111001011011101110010110111110110100011111101110111011000111001010010110001001011011100110101111100100000101110000001011110011101110101101001011111100011001100111001011011001011011101000101100100011101000011101011010110010111111010011110111111111100001101101001101101111000100110010010100000100000100110100100110010000100101100000101010101000111110101001111111011011111101101110010100110000010110111001101110011101100001111111001110110010010011101111100111111000010011010010011100000001010011111111110001111110110110110001001001111011001101110110101100011010001111110010001011010000011110101100110010010001110010100010000000010101010010111011010101011011000010010011010110101101011100100101110001100000111110010101100010100010100010000101111000011111000111000011110001110110000011000100110010101111111111100011010000010010010111111110000000010100100101101101010111011000001010000000101011010110011101000110011011001110101101001001101100010011110001101001001000011001101100001101100000101111000001110100001100001000100000000011010011001001011110011000011011000100011011001011100111111000011011010000100000011110110011011110111000001011101101110110101000101011111110000100000111111110011111111100010101001000011010001110111111011100010111001110110001000101010001001001111111110111001101001001000110000110100000000010010100001010111101001110001011101011000100000001001001011111000100011111000011010001110001111111110111100010001110101011101011100010100000011110000100011001100000100111000111011011101000110010011010011110010010000111010001011111111010110011100111011110100111000101100100111001001010001000100011110101100100010000000000110010111101101111101011001101111000000000100000101001011110000010011110011000000100111001001110001000010011111010100001001010010100111111010111001001110110101010010100010010100101100000001010010111101101000110100111110000000111110000011110011001101010111101010111010111010000000110010001011110101110110001000110111101110101101001001100001101110001000100000110100000100000111110010110110010011111100010011111001110111011010101010001011111110000000011110001010011001101010110010101100101100110010111001100100111101000100010100010010100000010010111101100111110101101111101001010110010001111111001101010111110101111001000111101001110100000111111010000111000010101001001101100111001111111111100011010111110001101000100001010000000001101101001110001000001110010000010110001001001101010111111001001010100011011111011000011001001101000110010101000010100001001000011111001101101111011111001000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    1000011010101011010111101011101111011100101001111111000101100100000011000100101001011001011110001101001100101001'

list1 = [int(i) for i in list((output[100:200]))]
y = vector(GF(2),list1) #值向量
list2 = []
for i in range(100):
    list2.append([int(j) for j in list(reversed(output[i:i+100]))])
x = matrix(GF(2),list2) #参数矩阵
mask = x.solve_right(y) #解方程x*mask=y
mask = ''.join([str(i) for i in list(mask)])
flag = int(mask,2)
print(flag)

得到结果

则flag为 flag{856137228707110492246853478448}