Quasilinear multiplication in the real Cayley--Dickson tower

Direct evaluation of the defining product in the real Cayley–Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge 1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge 3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge 16$ and than the uniform Cariow–Cariowa method for $N\ge 32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. 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 $N\ge 32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.