The knapsack cryptosystem at work

This note is an example of the knapsack crypto-system at work.So before you got through this note I recommend reading the main one . Click here

Suppose that a typical user of this crypto-system selects as the secret key the super-increasing sequence ${3,5,11,20,41}$ and the modulus $m = 85$ and the multiplier $a = 44$.Each member of the super-increasing sequence is multiplied by 44 and reduced modulo 85 to yield ${47,50,59,30,19}$.This is the encryption key the user submit to the public directory.

Suppose someone wants to send a plaintext message to the user such as "HELP US". First we convert it into a string of 1's and 0's $M = 00111 00100 01011 10100 10010$ The string is then broken up into blocks of digits,in the current case,blocks of length 5.Using the listed public keys to encrypt,the sender transforms the successive blocks into $47*0+50*0+59*1+30*1+19*1 = 108$ $47*0+50*0+59*1+30*0+19*0 = 59$ $47*0+50*1+59*0+30*1+19*1 = 99$ $47*0+50*1+59*1+30*1+19*1 = 158$ $47*1+50*0+59*1+30*0+19*0 = 106$ $47*1+50*0+59*0+30*1+19*0 = 77$

The transmitted ciphertext consists of the following series of positive integers ${108,59,99,158,106,777}$.

To read the message the legitimate receiver first solves the congruence $44x \equiv 1 \pmod{85}$,yielding $y \equiv 29 \pmod{85}$.Then each ciphertext number is multiplied by 29 and reduced modulo 85 to produce a super-increassing knapsack problem.

For example,$108$ is converted to $72$ as $108*29 \equiv 72 \pmod{85}$ and the corresponding knapsack problem is $3x_1 + 5x_2 + 11x_3 + 20x_4+41x_5 = 72$.

The procedure for solving super-increasing knapsack problems quickly yields the solution $x_1 = x_2 =0$,$x_3 = x_4=x_5 = 1$,in this way $00111$,the first block of the binary equivalent of the plaintext is obtained.