TY - CHAP

T1 - Finite field arithmetic

AU - Doche, Christophe

PY - 2006

Y1 - 2006

N2 - In this chapter, we are mainly interested in performance; see Section 2.3 for a theoretical presentation of finite fields. In the following, we consider three kinds of fields that are of great cryptographic importance, namely prime fields, extension fields of characteristic 2, and optimal extension fields. We will describe efficient methods for performing elementary operations, such as addition, multiplication, inversion, exponentiation, and square roots. The material that we give is implicitly more related to a software approach; see Chapter 26 for a presentation focused on hardware. Efficient finite field arithmetic is crucial in efficient elliptic or hyperelliptic curve cryptosystems and is the subject of abundant literature [JUN 1993, LINI 1997, SHP 1999]. See also the preliminary version of a book written by Shoup and available online [SHO], introducing basic concepts from computational number theory and algebra, and including all the necessary mathematical background.

AB - In this chapter, we are mainly interested in performance; see Section 2.3 for a theoretical presentation of finite fields. In the following, we consider three kinds of fields that are of great cryptographic importance, namely prime fields, extension fields of characteristic 2, and optimal extension fields. We will describe efficient methods for performing elementary operations, such as addition, multiplication, inversion, exponentiation, and square roots. The material that we give is implicitly more related to a software approach; see Chapter 26 for a presentation focused on hardware. Efficient finite field arithmetic is crucial in efficient elliptic or hyperelliptic curve cryptosystems and is the subject of abundant literature [JUN 1993, LINI 1997, SHP 1999]. See also the preliminary version of a book written by Shoup and available online [SHO], introducing basic concepts from computational number theory and algebra, and including all the necessary mathematical background.

KW - finite fields

KW - prime fields

KW - extension fields of characteristic 2

KW - optimal extension fields

UR - http://www.scopus.com/inward/record.url?scp=64049099956&partnerID=8YFLogxK

M3 - Chapter

SN - 9781584885184

T3 - Discrete mathematics and its applications

SP - 201

EP - 237

BT - Handbook of elliptic and hyperelliptic curve cryptography

PB - CRC Press, Taylor & Francis Group

CY - Boca Raton, Florida, USA

ER -