3 Commits

Author SHA1 Message Date
Daniel Lubarov
6072fab077
Implement a mul-add circuit in the ALU (#495)
* Implement a mul-add circuit in the ALU

The inputs are assumed to be `u32`s, while the output is encoded as four `u16 limbs`. Each output limb is range-checked.

So, our basic mul-add constraint looks like

    out_0 + 2^16 out_1 + 2^32 out_2 + 2^48 out_3 = in_1 * in_2 + in_3

The right hand side will never overflow, since `u32::MAX * u32::MAX + u32::MAX < |F|`. However, the left hand side could overflow, even though we know each limb is less than `2^16`.

For example, an operation like `0 * 0 + 0` could have two possible outputs, 0 and `|F|`, both of which would satisfy the constraint above. To prevent these non-canonical outputs, we need a comparison to enforce that `out < |F|`.

Thankfully, `F::MAX` has all zeros in its low 32 bits, so `x <= F::MAX` is equivalent to `x_lo == 0 || x_hi != u32::MAX`. `x_hi != u32::MAX` can be checked by showing that `u32::MAX - x_hi` has an inverse. If `x_hi != u32::MAX`, the prover provides this (purported) inverse in an advice column.

See @bobbinth's [post](https://hackmd.io/NC-yRmmtRQSvToTHb96e8Q#Checking-element-validity) for details. That post calls the purported inverse column `m`; I named it `canonical_inv` in this code.

* fix

* PR feedback

* naming
2022-02-21 00:39:04 -08:00
Jakub Nabaglo
83a572717e
Implement Poseidon in system_zero/permutation_unit (#459)
* Implement Poseidon in system_zero/permutation_unit

* Minor cleanup

* Daniel PR comments

* Update dependencies
2022-02-04 16:50:57 -08:00
Daniel Lubarov
c0ac79e2e1
Beginning of STARK implementation (#413)
* Beginning of STARK implementation

* PR feedback

* minor

* Suppress warnings for now
2022-01-26 00:09:29 -08:00