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

Please check out the visual layout of the circuit at: https://github.com/QED-it/halo2/blob/point-inequality-hash/point-inequality.png

[PaulLaux]

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