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 $N=1024$.
In this paper we investigate the maximum size of finite semigroups of rational $n \times n$ matrices, with the goal of shedding more light on their structure. Such semigroups provide a rich generalisation of transition monoids of unambiguous (and, in particular, deterministic) finite automata. While in general such semigroups can be arbitrarily large in terms of $n$, a classical result of Schützenberger from 1962 implies an upper bound of $2^{O(n^2 \log n)}$ for irreducible semigroups. A semigroup of rational matrices is called irreducible if the only subspaces of $\mathbb{Q}^n$ that are invariant for all matrices in the semigroup are $\mathbb{Q}^n$ and the subspace consisting only of the zero vector. Irreducible matrix semigroups can be viewed as the building blocks of general matrix semigroups, and as such play an important role in mathematics and computer science. From the point of view of automata theory, they can be seen as a generalisation of strongly connected weighted automata. Using a very different technique from that of Schützenberger, we improve the upper bound on the cardinality to $3^{n^2}$. This is the main result of the paper. The bound is in some sense tight, as we show that there exists, for every $n$, a finite irreducible semigroup with $3^{\lfloor n^2/4 \rfloor}$ rational matrices. Our main result also leads to an improvement of a bound, due to Almeida and Steinberg, on the mortality threshold of finite semigroups of rational matrices. The mortality threshold is a number $\ell$ such that if the zero matrix is in the semigroup, then the zero matrix can be written as a product of at most $\ell$ matrices from any subset that generates the semigroup.
John Graham, Soundes Marzougui, Miloš Pavlović, Vladimir Shpilrain
All traditional encryption schemes share a fundamental operational vulnerability: encrypted data must be decrypted before it can be used. Fully homomorphic encryption eliminates this vulnerability by enabling computation directly on encrypted data, but existing FHE schemes impose substantial practical constraints that limit deployment. This paper presents a formal security analysis of the Gribov-Kahrobaei-Shpilrain (GKS) private-key FHE scheme. GKS requires no bootstrapping or de-linearization, produces bit-exact decryption, supports native encrypted search, and scales to unbounded depth.
We prove two universality theorems for neural networks consisting of a single neuron. First we prove a strong form of the Bovdi--Geche universality conjecture on neurons over finite fields, where the activation function is a certain modification of the discrete logarithm, using the Weil bound on multiplicative character sums over finite fields. Then we show a generalisation of Aizenberg's universality theorem for complex binary neurons with a weight vector independent of the target function, positively answering another question in the literature. The proof of the latter uses an equidistribution theorem for the dilates of certain maps into tori.