By Gerasimos C. Meletiou, Arne Winterhof (auth.), Joachim von zur Gathen, José Luis Imaña, Çetin Kaya Koç (eds.)
This publication constitutes the refereed court cases of the second one overseas Workshop at the mathematics of Finite Fields, WAIFI 2008, held in Siena, Italy, in July 2008.
The sixteen revised complete papers provided have been rigorously reviewed and chosen from 34 submissions. The papers are prepared in topical sections on buildings in finite fields, effective finite box mathematics, effective implementation and architectures, category and building of mappings over finite fields, and codes and cryptography.
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Extra resources for Arithmetic of Finite Fields: 2nd International Workshop, WAIFI 2008 Siena, Italy, July 6-9, 2008 Proceedings
For aﬃne coordinates, I stands for the cost of a ﬁeld inversion. Table 1. Cost of point addition and doubling for various coordinate systems System Point addition Aﬃne (A) Homogeneous (H) Jacobian (J ) Chudnovsky (J c ) Modiﬁed Jacobian (J m ) 2M + S + I 12M + 2S 11M + 5S 10M + 4S 11M + 7S + 1c Point doubling (a4 = −3) 2M + 2S + I — 7M + 3S 5M + 6S + 1c 3M + 5S 1M + 8S + 1c 2M + 8S + 1c 4M + 5S 3M + 5S — When using projective coordinates, we see that Chudnovsky coordinates yield the faster point addition and that modiﬁed Jacobian coordinates yield the faster point doubling on any elliptic curve.
Note that, if not properly protected against, all left-to-right point multiplication methods (including the Montgomery ladder) are subject to the doubling attack. 5 Conclusion This paper presented an optimized implementation for inversion-free point multiplication on elliptic curves. In certain settings, the proposed implementation outperforms all such previously known methods without precomputation. Further, it scans the scalar from the right to left, which oﬀers a couple of additional advantages.
Possibilities, and for the N1 orbits of size p deﬁnes uniquely a balanced function from a set of qr elements to a set of q elements. M (qr, r, . . , r). 4 Degree of Transitive Balanced Boolean Functions When q = 2, n = pk with p prime >2 and f is a transitive balanced function on GF (2)n then from proposition 10 (f (0n ), f (1n )) is either (0,1) or (1,0). A balanced function has no term of degree n, its maximum degree can only be n − 1 and we have. Proposition 12. If n = pk with p prime > 2, then exactly half of the G transitive balanced functions from GF (2)n onto GF (2) are of maximum degree n − 1 and half of the G transitive balanced functions have algebraic degree strictly less than n − 1.
Arithmetic of Finite Fields: 2nd International Workshop, WAIFI 2008 Siena, Italy, July 6-9, 2008 Proceedings by Gerasimos C. Meletiou, Arne Winterhof (auth.), Joachim von zur Gathen, José Luis Imaña, Çetin Kaya Koç (eds.)