Need to work cryptography task

Learning Goal: I’m working on a computer science question and need an explanation and answer to help me learn.

Task

  1. A stream cipher uses a LFSR with three units. An attacker captures a ciphertext “0101001010011”. The attacker knows that the first 6 bits of the message are “110001.” Using this information, the attacker defeats this encryption, i.e., he is able to determine the initial seed values (S1, S2 and S3) and Keys (K1, K2 and K3). What are those values? (Points 25)

Calculation Hints:

Remember:

S 1 through S 6 can be computed by XORing 6 bit message with the first 6 bits of ciphertext.Then formulate the following equations for S 4 through S 6 :

S m+1= S m × K m + S m-1 × K m-1 + …… S 2 × K 2 + S 1 × K 1 mod 2

S m+2= S m+1 × K m + S m × K m-1 + …… S 3 × K 2 + S 2 × K 1 mod 2

S m+3= S m+2 × K m + S m+1 × K m-1 + …… S 4 × K 2 + S 3 × K 1 mod 2

Number of units in the shift register determines the value of m.

Many of you may not feel comfortable solving systems of equation. You can apply the following brute force method:

Check, all possible sets of values of K 1, K 2 and K 3 listed in the following table to find out which set satisfies all the above equations (i.e., makes both sides of the equations equal) :

K1 K2 K3
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
  1. An organization uses same stream cipher key multiple times. An attacker who knows that, captures two ciphertexts C1 and C2 and finds that,

C1⊕C2= 0000101000011101000000110001001100010000000101000000011000000011000010100000100100011011

The attacker also knows that the plaintexts are composed of only English capital letters without any punctuations or space, and that part of the plaintexts contains the word: “PRESIDENT”. He was able to recover both the plaintexts. Show, how. (Points 25)

Please see the attached screenshot

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