Hardware acceleration of the Tate pairing on a genus 2 hyperelliptic curve

2007 ◽  
Vol 53 (2-3) ◽  
pp. 85-98 ◽  
Author(s):  
Robert Ronan ◽  
Colm ó hÉigeartaigh ◽  
Colin Murphy ◽  
Michael Scott ◽  
Tim Kerins
Keyword(s):  
Hyperelliptic Curve ◽  
Hardware Acceleration ◽  
Tate Pairing ◽  
Genus 2

scholarly journals Cryptographic aspects of real hyperelliptic curves

2010 ◽  
Vol 47 (1) ◽  
pp. 31-65 ◽  
Author(s):  
Michael J. Jacobson ◽  
Renate Scheidler ◽  
Andreas Stein
Keyword(s):  
Hyperelliptic Curve ◽  
Discrete Logarithm ◽  
Mathematical Structure ◽  
Hyperelliptic Curves ◽  
Cryptographic Protocols ◽  
Discrete Logarithm Problem ◽  
Explicit Formulas ◽  
Genus 2

Abstract In this paper, we give an overview of cryptographic applications using real hyperelliptic curves. We review previously proposed cryptographic protocols and discuss the infrastructure of a real hyperelliptic curve, the mathematical structure underlying all these protocols. We then describe recent improvements to infrastructure arithmetic, including explicit formulas for divisor arithmetic in genus 2, and advances in solving the infrastructure discrete logarithm problem, whose presumed intractability is the basis of security for the related cryptographic protocols.

scholarly journals Flexible Prime-Field Genus 2 Hyperelliptic Curve Cryptography Processor with Low Power Consumption and Uniform Power Draw

ETRI Journal ◽  
2015 ◽  
Vol 37 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Hamid-Reza Ahmadi ◽  
Ali Afzali-Kusha ◽  
Massoud Pedram ◽  
Mahdi Mosaffa
Keyword(s):  
Power Consumption ◽  
Low Power ◽  
Hyperelliptic Curve ◽  
Low Power Consumption ◽  
Prime Field ◽  
Power Draw ◽  
Genus 2 ◽  
Hyperelliptic Curve Cryptography

scholarly journals Computing Hasse–Witt matrices of hyperelliptic curves in average polynomial time

2014 ◽  
Vol 17 (A) ◽  
pp. 257-273 ◽  
Author(s):  
David Harvey ◽  
Andrew V. Sutherland
Keyword(s):  
Polynomial Time ◽  
Efficient Algorithm ◽  
Hyperelliptic Curve ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Hyperelliptic Curves ◽  
Alternative Methods ◽  
Good Reduction ◽  
Genus 2 ◽  
Order Of Magnitude

AbstractWe present an efficient algorithm to compute the Hasse–Witt matrix of a hyperelliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C/\mathbb{Q}$ modulo all primes of good reduction up to a given bound $N$, based on the average polynomial-time algorithm recently proposed by the first author. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for $N = 2^{26}$.

Note on hyperelliptic surfaces and certain kummer surfaces

1936 ◽  
Vol 32 (3) ◽  
pp. 342-354 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Hyperelliptic Curve ◽  
Meromorphic Functions ◽  
Hyperelliptic Curves ◽  
Inversion Problem ◽  
Kummer Surface ◽  
Functions Of Two Variables ◽  
Hyperelliptic Surfaces ◽  
Genus 2 ◽  
Kummer Surfaces ◽  
Hyperelliptic Surface

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.

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