# Mixed Point Addition with Precomputation

The main computation of the MSM design is performed by a fully pipelined point adder. A mixed Twisted Edwards point adder computes the addition between a running sum in extended coordinate system and a point in affine coordinate system.

In our work, we implement a strongly unified mixed adder with `7M + 6A`, by exploiting some precomputation. This is a very big improvement over ```7M + 4S + 9A``` required for Jacobian coordinates, and a small but meaningful improvement over existing known methods for twisted edwards curves (`8M + 1*a + 7A`) and scaled twisted edwards curves (`7M + 1*k + 8A + 1*2`). Decreasing a multiplication might not seem like a lot, but when making FPGA designs operating at the edge at the device’s capabilities, every little counts in increasing performance!

While there is a mixed addition formula for addition in scaled twisted Edwards form with `7M + 8A`, that formula is not strongly unified. Using this implementation would have required us to carefully identify edge cases such as infinities and when the two points are identical (despite being numerically unequal). Using a strongly unified adder side-steps all of this entirely, which makes the MSM core a lot easier to reason about.

In the text that follows, points in the affine coordinate system consist of 3 elements \$(x,y,t)\$, where \$x * y = t\$. Points in the extended coordinate systems consist of 4 elements \$(x,y,z,t)\$, where \$x/z × y/z = t/z\$.

In both cases, \$t\$ is a redundant term meant make evaluation faster.

The nature of our adder’s workload looks something like computing a running sum, ie something like the following:

``````# ps is loaded at startup, and doesn't change across every evaluation batch,

acc = identity_element
for i in indices:
return acc
``````

We can make some observations here:

• The intermediate value of `acc` is never accessed
• `ps` is known ahead of time

The key idea in our precomputation optimization is to do some preprocessing in the affine points ahead of any evaluation and a single postprocessing on the accumulation result per MSM. The preprocessing is non-trivial, but can be computed ahead of time. The postprocessing is only executed once per bucket, which is negligible.

## The Precomputation Optimization

The exact proof that these formulae are equivalent to the vanilla mixed addition formulae after the transformations is beyond the scope of this document, but it should be a straightforward algebra exercise.

### Preprocessing on Affine Points

Ahead of any MSM evaluations, all affine points are transformed as follows:

\[(x_{new},y_{new},t_{new}) → ( (y-x)/2, (y+x)/2, 4dt )\]

Ahead of any MSM evaluations, all points are initialized to \$(2, 2, 4, 0)\$ (the rationale of this number will be elaborated below). Addition between a preprocessed affine point \$(x_{affine}, y_{affine}, t_{affine})\$ and the running sum \$(x_{running}, y_{running}, z_{running}, t_{running})\$ is defined as follows:

\$A = x_{running} × x_{affine}\$

\$B = y_{running} × y_{affine}\$

\$C = t_{running} × t_{affine}\$

\$D = z_{running}\$

\$E = B - A\$

\$F = D - C\$

\$G = D + C\$

\$H = B + A\$

\$I = E × F\$

\$J = G × H\$

\$t_{out} = E × H\$

\$z_{out} = F × G\$

\$x_{out} = J - I\$

\$y_{out} = J + I\$

### Post-Processing

Upon completion of bucket aggregation, the bucket sum results are post-processed using the following transformation:

\[(p, q, r, s) → ( (p-q)/4, (p+q)/4, r/4, s )\]

## Implementation Details

Our implementation supports a fully-pipelined mixed precompute adder (one that can accept input every cycle). Our codebase also supports a half-pipelined adder (one that can accept input every two cycles), but we did not use it as part of our final design.

The configuration of the adder can be specified via the a config type. The `slr_assignments` field in the config allows the user to specify which SLR every stage should be mapped to. Under the hood, this does two things:

• inserts a SLR{0,1,2} suffix to the instantiation name, which the user can use for pblocking in placement
• automatically insert registers between stages which are not in the same SLR. These registers are hierarchically created with SLR{0,1,2} suffix in their instantiation name so they can be pblocked.

In our implementation, 2 field multiplications in our adder are in SLR1, and 5 of them are in SLR2.

Some big benefits of using Hardcaml is that we can parameterize our design over these expressive configurations, and have them go through our regular tests on our own machines and Github actions. This gives us confidence that the design does not regress as we make config changes while conducting experiments for SLR assignments, which very rapidly increases productivity.

Relevant code: