# Handbook of Elliptic and Hyperelliptic Curve Cryptography

@inproceedings{Cohen2005HandbookOE, title={Handbook of Elliptic and Hyperelliptic Curve Cryptography}, author={Henri Cohen and Gerhard Frey and Roberto Maria Avanzi and Christophe Doche and Tanja Lange and Kim Nguyen and Frederik Vercauteren}, year={2005} }

Preface Introduction to Public-Key Cryptography Mathematical Background Algebraic Background Background on p-adic Numbers Background on Curves and Jacobians Varieties Over Special Fields Background on Pairings Background on Weil Descent Cohomological Background on Point Counting Elementary Arithmetic Exponentiation Integer Arithmetic Finite Field Arithmetic Arithmetic of p-adic Numbers Arithmetic of Curves Arithmetic of Elliptic Curves Arithmetic of Hyperelliptic Curves Arithmetic of Specialâ€¦Â

## 1,109 Citations

Empirical optimization of divisor arithmetic on hyperelliptic curves over F2m

- Mathematics, Computer ScienceAdv. Math. Commun.
- 2013

This paper describes how Nagao's methods, together with a sub-quadratic complexity partial extended Euclidean algorithm using the half-gcd algorithm can be applied to improve arithmetic in the degree zero divisor class group.

Arithmetic Units for the Elliptic Curve Cryptography with Concurrent Error Detection Capability

- Computer Science, Mathematics
- 2010

This thesis considers the elliptic curve cryptography over binary extension elds from two points of view, and designs concurrent error detection schemes for di erent Montgomery multipliers.

Constructing Hyperelliptic Covers for Elliptic Curves over Quadratic Extension Fields

- Mathematics, Computer ScienceACISP
- 2019

A method to generate genus 2 curves for which the point counting problems can be easily solved with efficient algorithms for elliptic curves.

Fast Multiple Point Multiplication on Elliptic Curves over Prime and Binary Fields using the Double-Base Number System

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2008

Three algorithms for multiple-point multiplication on elliptic curves over prime and binary fields are described, based on the representations of two scalars, as sums of mixed powers of 2 and 3.

Scalar multiplication on WeierstraÃŸ elliptic curves from Co-Z arithmetic

- Mathematics, Computer ScienceJournal of Cryptographic Engineering
- 2011

This paper describes efficient co-Z based versions of Montgomery ladder, Joyeâ€™s double-add algorithm, and certain signed-digit algorithms, as well as faster (X, Y)-only variants for left-to-right versions.

Comparative analysis of the scalar point multiplication algorithms in the NIST FIPS 186 elliptic curve cryptography

- Computer Science, MathematicsICCS-DE
- 2021

This paper analyzes the efficiency of the scalar multiplication on elliptic curves comparing Affine, Projective, Jacobian, Jacobi-Chudnovsky, and Modified Jacobian representations of an elliptic curve and shows that the Window method is the best providing lower execution time on considered coordinate systems.

Elliptic Curve Cryptography and Point Counting Algorithms

- Mathematics, Computer Science
- 2012

This chapter discussed the two point counting algorithms, Schoof algorithm and the SEA (Schoof-Elkies-Atkin) algorithm, which are part of the security of elliptic curve cryptography, and some similarities and the differences between these two algorithms.

Hyperelliptic Covers of Different Degree for Elliptic Curves

- Mathematics, Computer ScienceMathematical Problems in Engineering
- 2022

The construction of the cover map from the hyperelliptic curves to the elliptic curves is considered to convert point counting problems on hyperellIP curves to those on elliptIC curves and to use as a kind of cover attacks.

A Proposed Implementation of Elliptic Curve Exponentiation over Prime Field (Fp) in the Global Smart Cards

- Computer Science, Mathematics
- 2013

Mixed coordinate systems over prime field is proposed in the EC encryption and digital signature of the global smart cards.

FPGA Implementation of Various Elliptic Curve Pairings over Odd Characteristic Field with Non Supersingular Curves

- Mathematics, Computer ScienceIEICE Trans. Inf. Syst.
- 2016

An FPGA implementation that supports various parameter settings of pairings on non supersingular pairing-friendly curves for which Montgomery reduction, cyclic vector multiplication algorithm, projective coordinates, and Tate pairing have been combinatorially applied is shown.

## References

SHOWING 1-2 OF 2 REFERENCES

Handbook of Elliptic and Hyperelliptic Curve Cryptography

- Mathematics, Computer Science
- 2005

The introduction to Public-Key Cryptography explains the development of algorithms for computing Discrete Logarithms and their applications in Pairing-Based Cryptography and its applications in Fast Arithmetic Hardware Smart Cards.