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We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, [169][170][171][172][173][174][175][176][177][178][179][180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields.The algebraic complexity problem that is richest in underlying structure is the problem of fast polynomial multiplication. Among the problems reducible to this are fast multiplication of multiple-precision numbers, greatest common divisors in polynomial rings, Hankel (and, in particular, over any finite field) that one can achieve using variations of the fast Fourier transformation method is 0(n log n). The corresponding scheme for A = Z is far from simple. To study the optimal Zalgorithm one has to study their reductions mod p and m.m.c. algorithms of polynomial multiplication over finite fields, particularly over F2. Over finite fields the m.m.c. algorithms of polynomial multiplication are not, in general, given by the Toom-Cook scheme. For example, /.k(m, n) 2 m + n -1 for an arbitrary field k of scalars, but the inequality becomes equality only when the field k has at least m + n -2 elements (4). Better lower bounds on m.m.c. can be deduced using the theory of error-correcting linear codes. In refs. 7 and 8 it was proved that Fu2(n, n) > 3.52 n for large n [e.g., gz(n, n) > 3.52 n], using the upper bound (9). In this paper we explore various connections between m.m.c. algorithms for polynomial multiplication and multiplication in finite extensions of fields and optimal linear codes.First, we improve on lower bounds on m.m.c. over finite fields. Among bilinear algorithms that we study are m.m.c. algorithms for multiplication in commutative k-algebras without zero divisors. Then we use the connection between the theory of linear codes and algebraic curves over finite fields [Goppa codes (10)]. We present all the relevant information from the theory of algebraic curves over finite fields. Our algorithms of polynomial multiplication can be interpreted as interpolation methods on algebraic curves. As a corollary of our results, we prove that Ak(m, n) = 0(m + n) for an arbitrary finite field k. Moreover, for the multiplicative complexity ptk(N) of multiplication in a finite extension X over k we obtain a bound /uk(N) -2(1 + c/IkI"'2)[2K:kI comparing favorably with our lower bounds. Our results generalize to lower and upper bounds on m.m.c. of multiplication in other algebraic structures over finite fields including group algebras. Section

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