Memory-saving computation of the pairing final exponentiation on BN curves

2016 ◽  
Vol 8 (1) ◽  
Author(s):  
Sylvain Duquesne ◽  
Loubna Ghammam
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Multiplicative Group ◽  
Field Extension ◽  
Second Step ◽  
Tate Pairing ◽  
Important Constraint ◽  
Memory Resources ◽  
Pairing Based Cryptography ◽  
Pairing Computation

AbstractTate pairing computation is made of two steps. The first one, the Miller loop, is an exponentiation in the group of points of an elliptic curve. The second one, the final exponentiation, is an exponentiation in the multiplicative group of a large finite field extension. In this paper, we describe and improve efficient methods for computing the hardest part of this second step for the most popular curves in pairing-based cryptography, namely Barreto–Naehrig curves. We present the methods given in the literature and their complexities. However, the necessary memory resources are not always given whereas it is an important constraint in restricted environments for practical implementations. Therefore, we determine the memory resources required by these known methods and we present new variants which require less memory resources (up to 37 %). Moreover, some of these new variants are providing algorithms which are also more efficient than the original ones.

scholarly journals On Small Characteristic Algebraic Tori in Pairing-Based Cryptography

2006 ◽  
Vol 9 ◽  
pp. 64-85 ◽  
Author(s):  
R. Granger ◽  
D. Page ◽  
M. Stam
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Algebraic Tori ◽  
Simple Observation ◽  
Supersingular Elliptic Curves ◽  
Algebraic Torus ◽  
Pairing Based Cryptography

The value ot the late pairing on an elliptic curve over a finite field may be viewed as an element of an algebraic torus. Using this simple observation, we transfer techniques recently developed for torus-based cryptography to pairing-based cryptography, resulting in more efficient computations, and lower bandwidth requirements. To illustrate the efficacy of this approach, we apply the method to pairings on supersingular elliptic curves in characteristic three.

Constructing Elliptic Curve Including Subgroup with Low Hamming Order

2010 ◽  
Vol 113-116 ◽  
pp. 6-9
Author(s):  
Mao Cai Wang ◽  
Han Ping Hu ◽  
Guang Ming Dai ◽  
Lei Pen
Keyword(s):  
Elliptic Curve ◽  
Prime Order ◽  
Tate Pairing ◽  
Practical Applications ◽  
Effective Generation ◽  
Miller’S Algorithm ◽  
Pairing Computation

In practical applications of pairing-based cryptosystems, the efficiency of pairing computation is a crucial factor. Recently, there have been many improvements for the computation of Tate pairing, which focuses on the arithmetical operations under given elliptic curve. Based to the characteristics that Miller’s algorithm will be improved tremendous if there are subgroups with order of low hamming prime above the elliptic curve, an algorithm of generating primes of low hamming with weight 3 is given in this paper. Then, we present an effective generation method of elliptic curve, which enable it feasible that there is certain some subgroup of low hamming prime order. The improvement of paring computation is marked above the elliptic curve generating by our method.

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

scholarly journals The Pairing Computation on Edwards Curves

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hongfeng Wu ◽  
Liangze Li ◽  
Fan Zhang
Keyword(s):  
Geometric Interpretation ◽  
Tate Pairing ◽  
Quadric Surfaces ◽  
Edwards Curves ◽  
Explicit Formulae ◽  
Pairing Computation ◽  
Twisted Edwards Curves ◽  
Group Law

We propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves.

scholarly journals Classification of Elliptic Cubic Curves Over The Finite Field of Order Nineteen

2016 ◽  
Vol 13 (4) ◽  
pp. 846-852
Author(s):  
Baghdad Science Journal
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Maximum Size ◽  
Cubic Curves ◽  
Projective Equivalence

Plane cubics curves may be classified up to isomorphism or projective equivalence. In this paper, the inequivalent elliptic cubic curves which are non-singular plane cubic curves have been classified projectively over the finite field of order nineteen, and determined if they are complete or incomplete as arcs of degree three. Also, the maximum size of a complete elliptic curve that can be constructed from each incomplete elliptic curve are given.

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