Fast Finite Field Multiplication

• Serdar Süer Erdem
• Tuğrul Yanik
• Çetin Kaya Koç

Introduction

Finite fields are the most commonly used arithmetical structures in cryptography [14,16] and coding [3,19,21]. Many algorithms in cryptographic and coding applications are defined in terms of finite field arithmetic operations. The elliptic curve cryptosystems [17,11] and the Diffie-Hellman key exchange [8] algorithm are important examples of such cryptographic applications. Also, common error control codes such as Reed-Solomon and BCH codes are based on finite field theory [4,21].

An algebraic field consists of a set and two operations defined over this set. The real numbers, the rational numbers, and the complex numbers under addition and multiplication are examples of algebraic fields. In fact, algebraic fields are the generalization of these usual number systems as described below.
• One of the field operations satisfies the general properties of the usual addition. For this operation, an identity element exists and each element has an inverse. This identity element is...

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Authors and Affiliations

• Serdar Süer Erdem
• 1
• Tuğrul Yanik
• 2
• Çetin Kaya Koç
• 3
1. 1.Gebze Institute of TechnologyGebze
2. 2.Fatih UniversityIstanbul
3. 3.City University of Istanbul & University of California Santa BarbaraSanta Barbara