Modern computer algebra
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Finite field linear algebra subroutines
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
FFPACK: finite field linear algebra package
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
A computational introduction to number theory and algebra
A computational introduction to number theory and algebra
Efficient matrix rank computation with application to the study of strongly regular graphs
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Pseudo-Paley graphs and skew Hadamard difference sets from presemifields
Designs, Codes and Cryptography
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Faster polynomial multiplication via multipoint Kronecker substitution
Journal of Symbolic Computation
Generic GF(2m) arithmetic in software and its application to ECC
ACISP'03 Proceedings of the 8th Australasian conference on Information security and privacy
The M4RIE library for dense linear algebra over small fields with even characteristic
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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We present algorithms to perform modular polynomial multiplication or a modular dot product efficiently in a single machine word. We use a combination of techniques. Polynomials are packed into integers by Kronecker substitution; several modular operations are performed at once with machine integer or floating point arithmetic; normalization of modular images is avoided when possible; some conversions back to polynomial coefficients are avoided; the coefficients are recovered efficiently by preparing them before conversion. We discuss precisely the required control on sizes and degrees. We then present applications to polynomial multiplication, prime field linear algebra and small extension field arithmetic, where these techniques lead to practical gains of quite large constant factors.