 
Summary: Theoretical Computer Science 58 (1988) 1756
NorthHolland
CLASSIFICATION OF ALL THE MINIMAL BILINEAR
ALGORITHMS FOR COMPUTING THE COEFFICIENTS OF
THE PRODUCT OF TWO POLYNOMIALS MODULO A
POLYNOMIAL, PART I: THE ALGEBRA G[u]/< Q(u)`>, 1> 1
Amir AVERBUCH
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A., and TelAviv University,
Ramat Aviv, 69978 TelAviv, Israel
Zvi GALIL
Columbia University, Morningside Heights, NY 10027, U.S.A., and TelAviv University, Ramat
Aviv, 69978 TelAviv, Israel
Shmuel WINOGRAD
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A.
Abstract. In this paper we will classify all the minimal bilinear algorithms for computing the
coefficients of (Cyld x,u')(~:`~~ y,u') mod Q(U)' where deg Q(u) = j, jl = n and Q(U) is irreducible.
The case where I = 1 was studied in [ 11. For I > 1 the main results are that we have to distinguish
between two cases: j > 1 and j = 1. The first case is discussed here while the second is classified
in [4]. For j > 1 it is shown that up to equivalence every minimal (2n  1 multiplications) bilinear
algorithm for computing the coefficients of (x:1; x,u')(C:`I,; y,u') mod Q(u)' is done by first
