[Back] This is a simple abstraction of homomorphic cipher for a decision if Bob is older than Alice. In this case we will define Bob and Alice's age will be defined in two bit binary values: 00 (0 - 10 years), 01 (10 - 20 years), 10 (20 - 30 years), and 11 (over 30 years):

## Simple Homomorphic Cipher for checking if Bob older |

## Theory

The code uses the DGHV (Dijk, Gentry, Halevi and Vaikuntanathan) scheme [paper] and which is a fully homomorphic encryption scheme [1].

First we get a random number (p) and two random numbers (q and r), and then encrypt a single bit (m) with:

\(c = p \times q + 2 \times r +m \)

In this case p will be our secret key.

If 2r is smaller than p/2, we cipher with mod p we get:

\(d = (c \mod p) \mod 2\)

We have basically added noise to the value with the q and r values.

With this function we can add (XOR) and multiply (AND).

In this example we will create a True Table for Bob (\(b_0b_1\) and Alice (\(a_1a_0\)). In this case we will return a 1 if Bob is older, and a 0 is he is not:

The logic is then:

\(Z_0=\bar{a_1} {b_1} + \bar{a_1} \bar{a_0} b_0 + \bar{a_0} b_1 b_0\)

## Code

The following is an outline:

import sys from random import randint a_0 = 0 a_1 = 0 b_0 = 1 b_1 = 0 def inv(val): return(val ^ 1) r1 =randint(1, 5) r2= randint(1, 5) r3 =randint(1, 5) r4 =randint(1, 5) q1 =randint(50000, 60000) q2 =randint(50000, 60000) q3 =randint(50000, 60000) q4 =randint(50000, 60000) p =randint(10000, 20000) c_bit_a_0 = q1 * p + 2*r1 +a_0 c_bit_a_1 = q2 * p + 2*r2 +a_1 c_bit_b_0 = q3 * p + 2*r3 +b_0 c_bit_b_1= q4 * p + 2*r4 +b_1 # Truth table cipher_older = inv(c_bit_a_1)*c_bit_b_1 + inv(c_bit_a_1)*inv(c_bit_a_0)*c_bit_b_0 + inv(c_bit_a_0)*c_bit_b_1*c_bit_b_0 print "r values:\t",r1,r2,r3,r4 print "q values:\t",q1,q2,q3,q4 print "p value:\t",p print "\nInput bits" print "---------------" print "a_1:\t",a_1, "a_0:\t",a_0 print "b_1:\t",b_1, "b_0:\t",b_0 print "\nCipherbits" print "---------------" print "a:\t",c_bit_a_0, "b:\t",c_bit_a_1 print "c:\t",c_bit_b_1, "d:\t",c_bit_b_1 print "\nCipher result" print "---------------" print "Result:\t",cipher_older #decrypt result = (cipher_older % p) % 2 print "\nResults" print result if (result==1): print "Bob is older than Alice" else: print "Bob is not older than Alice"

## Presentation

Here is a presentation on the method used:

## Reference

[1] M. van Dijk, C. Gentry, S. Halevi and V. Vaikuntanathan, "Fully Homomorphic Encryption over the Integers". Proceedings of Eurocrypt 2010.