Parallel algorithms for algebraic problems
SIAM Journal on Computing
Representations and parallel computations for rational functions
SIAM Journal on Computing
Subresultants and Reduced Polynomial Remainder Sequences
Journal of the ACM (JACM)
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
On Euclid's Algorithm and the Theory of Subresultants
Journal of the ACM (JACM)
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
On rank properties of Toeplitz matrices over finite fields
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Designs, Codes and Cryptography
Early termination over small fields
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Fast rational function reconstruction
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Relatively prime polynomials and nonsingular Hankel matrices over finite fields
Journal of Combinatorial Theory Series A
Multiple GCDs. probabilistic analysis of the plain algorithm
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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This paper analyzes the Euclidean algorithm and some variants of it for computingthe greatest common divisor of two univariate polynomials over a finite field. The minimum, maximum and average number of arithmetic operations both on polynomials and in the ground field are derived. We consider five different algorithms to compute gcd(A"1, A"2) where A"1, A"2@?Z"2[x] have degrees m=n=0. Compared with the classical Euclidean algorithm that needs on average 1/2n+1 polynomial divisions, two algorithms involving divisions need on average 1/3n+O(1) and 1/4n+O(1) polynomial divisions; two other algorithms use an average of 1/2m+1/3n+O(1) and 1/4m+2/9n+O(1) polynomial substractions and no divisions.