2009, pp 105-124

Efficient Unified Arithmetic for Hardware Cryptography

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The basic arithmetic operations (i.e., addition, multiplication, and inversion) in finite fields, \(GF(q)\) , where \(q = p^k\) and p is a prime integer, have several applications in cryptography, such as RSA algorithm, Diffie-Hellman key exchange algorithm [1], the US federal Digital Signature Standard [2], elliptic curve cryptography [3,4], and also recently identity-based cryptography [5, 6]. Most popular finite fields that are heavily used in cryptographic applications due to elliptic curve-based schemes are prime fields \(GF(p)\) and binary extension fields \(GF(2^n)\). Recently, identity-based cryptography based on pairing operations defined over elliptic curve points has stimulated a significant level of interest in the arithmetic of ternary extension fields, \(GF(3^n)\).

Even though the aforementioned three popular finite fields are dissimilar mathematical structures, their elements are represented using similar data structures inside the digital circuits and computers. F ...