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The F4-algorithm for Euclidean rings

100%

Sadiq A.

tom 8

nr 6

1156-1159

In this short note, we extend Faugére’s F4-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F4-algorithm is based on the simultaneous reduction of several polynomials.

Simple Verification of Completeness of Two Addition Formulas on Twisted Edwards Curves

80%

Dryło R. , Kijko T.

tom Vol. 66, No. 3

459--464

Daniel Bernstein and Tanja Lange [9] proved that two given addition formulas on twisted Edwards elliptic curves ax² + y² = 1 + dxy are complete (i.e. the sum of any two points on a curve can be computed using one of these formulas). In this paper we give simple verification of completeness of these formulas using a program written in Magma, which is based on the fact that completeness means that some systems of polynomial equations have no solutions. This method may also be useful to verify completeness of additions formulas on other models of elliptic curves.