# Results on complexity of multiplication over finite fields

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2009
Cenk, Murat
Let n and l be positive integers and f (x) be an irreducible polynomial over Fq such that ldeg( f (x)) < 2n - 1, where q is 2 or 3. We obtain an effective upper bound for the multiplication complexity of n-term polynomials modulo f (x)^l. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for polynomial multiplication over Fq. We give improved formulae to multiply polynomials of small degree over Fq. In particular we improve the best known multiplication complexities over Fq in the literature in some cases. Moreover, we present a method for multiplication in finite fields improving finite field multiplication complexity \muq(n) for certain values of q and n. We use local expansions, the lengths of which are further parameters that can be used to optimize the bounds on the bilinear complexity, instead of evaluation into residue class field. We show that we obtain improved bounds for multiplication in Fq^n for certain values of q and n where 2 <= n < =18 and q = 2, 3, 4.

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Citation Formats
M. Cenk, “Results on complexity of multiplication over finite fields,” Ph.D. - Doctoral Program, Middle East Technical University, 2009.