Another look at square roots (and other less common operations) in fields of even characteristic

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Abstract

We discuss a family of irreducible polynomials that can be used to speed up square root extraction in fields of characteristic two. They generalize trinomials discussed by Fong et al. [20]. We call such polynomials square root friendly. The main application is to point halving methods for elliptic curves (and to a lesser extent also divisor halving methods for hyperelliptic curves and pairing computations). We note the existence of square root friendly trinomials of a given degree when we already know that an irreducible trinomial of the same degree exists, and formulate a conjecture on the degrees of the terms of square root friendly polynomials. Following similar results by Bluher. we also give a partial result that goes in the direction of the conjecture. We also discuss how to improve the speed of solving quadratic equations. The increase in the time required to perform modular reduction is marginal and does not affect performance adversely. Estimates confirm that the new polynomials mantain their promises. Point halving gets a speed-up of 20% and scalar multiplication is improved by at least 11%.

Original languageEnglish
Title of host publicationSelected Areas in Cryptography - 14th International Workshop, SAC 2007, Revised Selected Papers
PublisherSpringer Verlag
Pages138-154
Number of pages17
ISBN (Print)3540773592, 9783540773597
DOIs
StatePublished - 2007
Externally publishedYes
Event14th International Workshop on Selected Areas in Cryptography, SAC 2007 - Ottawa, Canada
Duration: 16 Aug 200717 Aug 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4876 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference14th International Workshop on Selected Areas in Cryptography, SAC 2007
Country/TerritoryCanada
CityOttawa
Period16/08/0717/08/07

Keywords

  • Binary fields
  • Point halving
  • Polynomial basis
  • Quadratic equations
  • Square root extraction
  • Trace computation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science

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