# Performance Scaling

## 8 parallel cores

The 4-step algorithm provides an obvious way to scale up performance. A large transform is broken up into thousands of smaller NTT transforms which may be performed in parallel.

This is the approach taken in our core. The first level of scaling is implemented in the parallel cores module which conceptually groups 8 NTT cores together. The actual design is slightly more optimal in that it shares a single controller module for all 8 data paths and related memories.

The grouping of 8 cores was chosen as this matches our memory bus width. Each core has a 64-bit input and output bus for loading and storing coefficients. $8 x 64 = 512$, which is the required width.

## Multi-Parallel Cores

The multi-parallel cores module instantiates multiple parallel cores, further increasing parallelism.

Each block of 8 parallel cores shares the memory bus at this level of the design so it must be decoded/multiplexed into a unified address space.

A simplification of the design made here is to only allow scaling at powers of 2 to simplify internal address decoding. This requirement limits our performance a little as we could match memory bandwidth or area constraints more accurately if we could scale up arbitrarily.

## 4-step controller

The 4-step controller module sequences a single pass of the 4-step algorithm. For a transform of size $2^24$ we must perform $2^12 = 4096$ NTTs. With, say, 32 cores we need to iterate $4096/32 = 128$ times per pass.

This module controls that iteration and sequences the memory access steps in parallel with the NTT transforms.

The overall performance of the design, assuming memory can be accessed quickly enough, is bound by the number of required iterations plus an initial memory load and final memory store operation.