Quasilinear multiplication in the Cayley-Dickson tower
Harrison Lemley
Direct evaluation of the defining product of the Cayley-Dickson algebra $A_n$ of rank $N:=2^n$ over a unital commutative ring $R$ in which $2$ is invertible requires a quadratic number of arithmetic operations. This paper gives a uniform bilinear algorithm for multiplication in this algebra costing $O(N \log N)$ arithmetic operations in $R$ and $O(N)$ auxiliary storage when the doubling parameters of $A_n$ are units whose square roots lie in a fixed commutative coefficient algebra $S$ finitely generated as an $R$-module and admitting an $R$-linear map $π\colon S \to R$ with $π(1) = 1$. In particular, over $\mathbb{R}$ and every finite field $\mathbb{F}_{q}$ of odd characteristic, a quadratic field extension contains a square root of every possible doubling parameter, so the algorithm applies to every Cayley-Dickson tower over these fields, and the tensor rank of multiplication in $A_n$ over them is at most $(9n-21)2^n+28$ when $n\ge2$. When the top doubling parameter is a non-square, as in the standard real tower, the bound is $(9n-15)2^{n-1}+10$. To the author's knowledge, the algorithm presented in this paper is the first uniform subquadratic algorithm for multiplication in $A_n$. When $R=\mathbb{R}$ with every doubling parameter equal to $-1$, the real quasilinear schedule uses fewer total arithmetic operations than both direct multiplication and a uniform algorithm of Cariow-Cariowa for $n\ge4$. An implementation for the standard real tower is available in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface. In single-core benchmarks against direct multiplication and the uniform Cariow-Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was over $17$ times faster than direct multiplication at $N=1024$.
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