# Optimizing Multi-Scalar Multiplication (MSM): Learning from ZPRIZE
The following algorithms and techniques are used by the top-2 performers of ZPrize 2022 MSM-WSAM track.
1. [Pippenger Algorithm / Bucket Method](/lNOsNGikQgO0hLjYvH6HJA)
2. [Batch Addition](/1mpavmFmQNWrahBi8mHBjQ)
3. [Signed Bucket Indexes](/jNXoVmBaSSmE1Z9zgvRW_w)
4. [GLV Decomposition](/VnWyAPOZRFeCcHg-aVzxpQ)
5. [Montgomery Multiplication](/lF8qDQZnR0quXhXJ1TkC5Q)
6. [30-bit Limbs of Multi-Precision Integers](/HKHbMzxIQmOQUFNl7CF5sg) (WASM Specific)
For more information, checkout
* [`arkmsm`](https://github.com/snarkify/arkmsm): our open source implementation of the optimization techniques above based on `arkworks`. We implemented the Pippenger algorithm with batch addition, signed bucket indexes and glv decomposition, while using montgomery multiplications implemented in `arkworks`.
* My [ZKSummit9 talk on `arkmsm` optimizations](https://youtu.be/j8f6phMp-g4?feature=shared), Lisbon, April 2023.
* The `ark-msm` crate on [crates.io](https://crates.io/crates/ark-msm)
###### tags: `msm` `zkp` `public` `arkmsm`

If we treat bucket indexes as signed integers, we are able to saves one bit in bucket index encoding, and therefore reduce number of buckets and bucket memory usage by half. The method is denoted as the signed-bucket-index method, and also known as NAF method. This method was reported in [1] and implemented by top-2 performers of the zPrize MSM-WASM track.

10/25/2023Problem Give $n$ scalars ${k_i}$ and $n$ EC points ${P_i}$, calculate $P$ such that $$P=\sum_{i=0}^n k_i P_i$$ The Pippenger / Bucket Algorithm Step 1: partition scalars into windows Let's first partition each scalar into $m$ windows each has $w$ bits, then $$k_i = k_{i,0} + k_{i,1} 2^{w} + ... + k_{i,(m-1)} 2^{(m-1)w}$$ You can think each scalar $k_i$ as a bignum and representing it as a multi-precision integer with limb size $w$.

10/1/2023GLV Decomposition is an efficient way to calculate point scaling $kP$. This method was originally published by Gallant, Lambert and Vanstone in 2001 [1], and was later integrated into the Bitcoin core code [2]. However, GLV Decomposition was not enabled in Bitcoin until a patent around the GLV method expired in 2020 [3]. A textbook introduction of enomorphisms of elliptic curves can be found in [4]. Problem Given scalar $k$ and EC point $P$, caculate $kP$. Property of Enomorphism Consider the elliptic curve over field $\mathbb{F}_p$ $$E: y^2 = x^3 + c\

3/29/2023Algorithm Consider prime field $\mathbb{F}_p$, select power of two $2^w$ such that $R=2^w > p$, we know that mod by R can be computed by bit shifting. Montgomery Form For $x \in \mathbb{F}_p$, define Montgomery form of $x$ as, $$\bar{x} = xR \pmod p$$ Transform To Montgomery Form Transforming $x$ to Montgomery form can be done by left-shift $x$ by $w$ and reduce modulo $p$. In practice, $x$ and $y$ are transformed to Montgomery form at the beginning of a computation, and transformed back at the end.

2/27/2023
Published on ** HackMD**

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