A Lower Bound on the Complexity of Division in Finite Extension Fields and Inversion in Quadratic Alternative Algebras

Abstract

Let k be a field. The author continues the work of Lickteig [SIAM J. Comput., 16 (1987), pp. 278–311 ] and extends it in two directions: (1) A lower bound on the Ostrowski complexity of division in a finite extension field $A \supset k:L_{{\text{DIV}}} \geqq 3[A:k] + \log _2 [B:k] - 2$, where $B \subset A$ is some simple extension of k of maximal degree. The proof combines the technique of adjoining intermediate results to the ground field [T. Lickteig, op, cit.] and lower bound criteria for approximative complexity recently obtained by Griesser [Theoret. Comput. Sci., 46 (1986), pp. 329–338].(2) An optimal lower bound for the complexity of inversion in a quadratic alternative algebra A of dimension greater than or equal to $2,A \ne k \times k:L_{{\text{INV}}} = 2\dim A - {\text{index of the norm}}$.