Searching for Best Karatsuba Recurrences
Efficient circuits for multiplication of binary polynomials use what are known as Karatsuba recurrences. These methods divide the polynomials of size (i.e. number of terms) \(k \cdot n\) into k pieces of size n. Multiplication is performed by treating the factors as degree-\((k-1)\) polynomials, with multiplication of the pieces of size n done recursively. This yields recurrences of the form \( M(k n) \le \alpha M(n) + \beta n + \gamma ,\) where M(t) is the number of binary operations necessary and sufficient for multiplying two binary polynomials with t terms each. Efficiently determining the smallest achievable values of (in order) \(\alpha , \beta , \gamma \) is an unsolved problem. We describe a search method that yields improvements to the best known Karatsuba recurrences for k = 6, 7 and 8. This yields improvements on the size of circuits for multiplication of binary polynomials in a range of practical interest.
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