Point inequality hash

[hridambasu]

Halo2 Example: Prove knowledge of a point (x, y) that is NOT equal to reference point (x0, y0)
This circuit demonstrates proving that a witness point (x, y) is different from

• a public reference point (x0, y0). The inequality is proven by showing that
• either x != x0 OR y != y0 (or both).

We solve this by computing the difference of the points

• dx = (x - x0), dy = (y - y0),
  and then choosing a random point r which is computed by taking the Hash of the transcript
  of the protocol so far and then checking whether a random linear combination
  of the differences of the values equates to zero or not.
• c = dx + r.dy
  The Hash can be a Poseidon hash or a Blake2s Hash

The circuit looks like:

Row | I0 | A0 | A1 | F0 | F1 | F2 | S0 | S1 | S2 | S3
----+--------------------------------------------------------------------+--------------------------------------------------------------------+--------------------------------------------------------------------+-----+----+----+----+----+----+---
0 | 0x5 | 0xa | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
1 | 0x7 | 0xc | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
2 | 0x2bcecee922cd968d7c3826eae86f127bb8e7229646410618eabf4e9772f56d42 | 0x5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
3 | 0 | 0x7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
4 | 0 | 0xa | 0x5 | 0x2 | 0 | 0 | 0 | 1 | 0 | 0
5 | 0 | 0x5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
6 | 0 | 0xc | 0x7 | 0x2 | 0 | 0 | 0 | 1 | 0 | 0
7 | 0 | 0x5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
8 | 0 | 0x2bcecee922cd968d7c3826eae86f127bb8e7229646410618eabf4e9772f56d42 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
9 | 0 | 0x2bcecee922cd968d7c3826eae86f127bb8e7229646410618eabf4e9772f56d42 | 0x5 | 0 | 1 | 0 | 0 | 0 | 1 | 0
10 | 0 | 0x1b0a0a8dae03f0c36d18c2968a2b5c6a35afe1fb435e3329ca34f62e3ecb2247 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
11 | 0 | 0x5 | 0x1b0a0a8dae03f0c36d18c2968a2b5c6a35afe1fb435e3329ca34f62e3ecb2247 | 1 | 0 | 0 | 1 | 0 | 0 | 0
12 | 0 | 0x1b0a0a8dae03f0c36d18c2968a2b5c6a35afe1fb435e3329ca34f62e3ecb224c | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
13 | 0 | 0x1b0a0a8dae03f0c36d18c2968a2b5c6a35afe1fb435e3329ca34f62e3ecb224c | 0x2d0ae087001c8a40106b757811460967fa234f183acde4f79b1bb7a879275813 | 0 | 0 | 1 | 0 | 0 | 0 | 1