## Abstract

In this paper we classify all the minimal bilinear algorithms for computing the coefficients of (Σ^{n-1}_{i=0} x_{i}u^{i}) (Σ^{n-1}_{i=0} y_{i}u^{i}) mod Q(u)^{l} where deg Q(u)=j, jl=n and Q(u) is irreducible (over G) is studied. The case where l = 1 was studied in [8]. For l > 1 the main results are that we have to distinguish between two cases: j > 1 and j = 1. The case where j > 1 was studied in [1]. For j = 1 it is shown that up to equivalence, every minimal (2n - 1 multiplications) bilinear algorithm for computing the coefficients of (Σ^{n-1}_{i=0} x_{i}u^{i}) (Σ^{n-1}_{i=0} y_{i}u^{i}) mod u^{n} is done either by first computing the coefficients of (Σ^{n-1}_{i=0} x_{i}u^{i}) (Σ^{n-1}_{i=0} y_{i}u^{i}) and then reducing them modulo u^{n} or by first computing the coefficients (Σ^{n-2}_{i=0} x_{i}u^{i}) (Σ^{n-1}_{i=0} y_{i}u^{i}) and then reducing them modulo u^{n} and adding x_{n}-1y_{0}u^{n-1} or by first computing the coefficients (Σ^{n-2}_{i=0} x_{i}u^{i}) (Σ^{n-2}_{i=0} y_{i}u^{i}) and then reducing them modulo u^{n} and adding (x_{n}-1y_{0} + x_{0}y_{n}-1)u^{n}-1.

Original language | English |
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Pages (from-to) | 143-203 |

Number of pages | 61 |

Journal | Theoretical Computer Science |

Volume | 86 |

Issue number | 2 |

DOIs | |

State | Published - 2 Sep 1991 |

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