Homomorphic Encryption for Gray Codes

When I studied electronic engineering, there was one code that was so simple to implement but could minimise the number of changes that we have in a circuit: the Gray Code.

For the Gray code, we make sure there is only one bit change for an incremental count. Thus for our binary code, if we go from 1 (0001) to 2 (0010), there is be two transistions of bits. But with a Gray code we can represent 1 by 0001 and 2 by 0011, and so there is only one transition of the bits. The truth table is then:

Decimal       Binary     Gray
----------------------------
0             0000 	 0000
1             0001 	 0001
2             0010 	 0011
3             0011  	 0010
4             0100 	 0110
5             0101 	 0111
6             0110 	 0101
7             0111 	 0100
8             1000 	 1100
9             1001 	 1101
10            1010 	 1111
11            1011 	 1110
12            1100 	 1010
13            1101 	 1011
14            1110 	 1001
15            1111 	 1000

The circuit for this is then:

Now, let’s implement this with homomorphic encryption. Overall, FHEW [1] uses a LWE (Learning With Error) method that provides fully homomorphic…