Multi-bit public key encryption with Learning With Errors (LWE)

Learning With Errors (LWE) is a quantum robust method of cryptography. Initially we create a secret value (s) and which is our private key. We then create a public key which is based on random numbers (A), and then generate another set of numbers (B) which is based on A,s and random errors e. In this case we will show how a 4-bit value can be encrypted. In this case we will convert an integer to a 4-bit value, and then cipher each of the bits. This is achieved by generating a public key, and then sampling the public key for each of the bits.

This is a method defined by Oded Regev in 2005 [here] and is known as LWE. First we select a random series of values for our public key (A). For example, let’s select 20 random values from 0 to 100:

[80, 86, 19, 62, 2, 83, 25, 47, 20, 58, 45, 15, 30, 68, 4, 13, 8, 6, 42, 92]

Next we add create a list (B) and were the elements are Bi=Ais+ei(modq), and where s is a secret value, and e is a list of small random values (the error values). If we take a prime number (q) of 97, and an error array (e) of:

[3, 3, 4, 1, 3, 3, 4, 4, 1, 4, 3, 3, 2, 2, 3, 2, 4, 4, 1, 3]

we generate a list (B) of:

[15, 45, 2, 20, 13, 30, 32, 45, 4, 3, 34, 78, 55, 51, 23, 67, 44, 34, 17, 75]