Fast Bilinear Maps from the Tate-Lichtenbaum Pairing on Hyperelliptic Curves

2006 ◽  
pp. 466-479 ◽  
Author(s):  
Gerhard Frey ◽  
Tanja Lange
Keyword(s):  
Hyperelliptic Curves ◽  
Bilinear Maps

scholarly journals Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Jennifer S. Balakrishnan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Hyperelliptic Curves ◽  
Open Book ◽  
Line Integral ◽  
Book Series ◽  
Numerical Examples ◽  
Lecture Notes ◽  
Integration Algorithms ◽  
Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

scholarly journals Integral points on hyperelliptic curves

2008 ◽  
Vol 2 (8) ◽  
pp. 859-885 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek ◽  
Michael Stoll ◽  
Szabolcs Tengely
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

scholarly journals A Novel Construction of Constrained Verifiable Random Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
Muhua Liu ◽  
Ping Zhang ◽  
Qingtao Wu
Keyword(s):  
Random Function ◽  
Secret Key ◽  
Bilinear Maps ◽  
Random Functions ◽  
Compute Function ◽  
Multilinear Maps ◽  
Verifiable Random Functions ◽  
Selective Security

Constrained verifiable random functions (VRFs) were introduced by Fuchsbauer. In a constrained VRF, one can drive a constrained key skS from the master secret key sk, where S is a subset of the domain. Using the constrained key skS, one can compute function values at points which are not in the set S. The security of constrained VRFs requires that the VRFs’ output should be indistinguishable from a random value in the range. They showed how to construct constrained VRFs for the bit-fixing class and the circuit constrained class based on multilinear maps. Their construction can only achieve selective security where an attacker must declare which point he will attack at the beginning of experiment. In this work, we propose a novel construction for constrained verifiable random function from bilinear maps and prove that it satisfies a new security definition which is stronger than the selective security. We call it semiadaptive security where the attacker is allowed to make the evaluation queries before it outputs the challenge point. It can immediately get that if a scheme satisfied semiadaptive security, and it must satisfy selective security.

The surfaces whose prime-sections contain a

1934 ◽  
Vol 30 (2) ◽  
pp. 170-177 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Hyperelliptic Curves ◽  
Ruled Surfaces ◽  
Minimum Order ◽  
Rational Surfaces ◽  
Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

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