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.
10/25/2023The following algorithms and techniques are used by the top-2 performers of ZPrize 2022 MSM-WSAM 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\
3/29/2023or
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