Arithmetic of Finite Fields: 2nd International Workshop, by Gerasimos C. Meletiou, Arne Winterhof (auth.), Joachim von

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 foreign Workshop at the mathematics of Finite Fields, WAIFI 2008, held in Siena, Italy, in July 2008.

The sixteen revised complete papers offered have been conscientiously reviewed and chosen from 34 submissions. The papers are geared up in topical sections on buildings in finite fields, effective finite box mathematics, effective implementation and architectures, class and building of mappings over finite fields, and codes and cryptography.

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Extra info for Arithmetic of Finite Fields: 2nd International Workshop, WAIFI 2008 Siena, Italy, July 6-9, 2008 Proceedings

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2. 3. 4. Compute Compute Compute Compute the the the the exponentiation Ar−1 in GF (pm ), where r = product Ar = (Ar−1 ) · A; inversion (Ar )−1 in GF (p); product Ar−1 · (Ar )−1 = A−1 . pm −1 p−1 ; For the particular choice of pm − 1 , p−1 Ar belongs to the ground field GF (p) [13]. This allows the inversion in step 3 to be computed in GF (p) instead of the larger field GF (pm ). For the exponentiation Ar−1 in step 1, the exponent r − 1 is expanded as follows r= r−1= pm − 1 − 1 = pm−1 + pm−2 + . .

If G is a transitive group and n = pk with p prime then for any x = (an ), we have: 1. p divides |xG | 2. n divides |xG |if there exists b such that gcd(p, wb(x)) = 1. 3 The n-i Transitive Functions The set of symmetric functions is not always a proper subset of the partially symmetric functions sets, due to some Pigeon Hole principle as inputs belong to the finite set Eq . Proposition 1. e. Gf the symmetry group of f is n-i transitive) if and only if f is a symmetric function from Eqn onto Em .

When the cost of point inversion is negligible, it is advantageous to input the NAF representation of k, k = i=0 ki 2i with ki ∈ {−1, 0, 1} and ki · ki+1 = 0, and to adapt the scalar multiplication method accordingly. For example, in Algorithm 2, Line 5, R1 is added when ki = 1 and R1 is subtracted when ki = −1. This strategy reduces the average number of point additions in the left-to-right binary method from ( − 1)/2 to /3. Fast Point Multiplication on Elliptic Curves without Precomputation 4 41 Fast Right-to-Left Point Multiplication In this section, we optimize as much as possible the binary right-to-left method for point multiplication on elliptic curves over fields K of characteristic = 2, 3.

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