Efficient Unified Arithmetic for Hardware Cryptography

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. Furthermore, similarity of algorithms for basic arithmetic operations in these fields allows a unified module design. For example, the steps of the original Montgomery multiplication algorithm [7], which is one of the most efficient methods for multiplication in finite fields, GF (p) and rings slightly differ from those of the Montgomery multiplication algorithm for binary extension fields, GF (2 n) given in [8]. In addition, it is almost straightforward to extend the Montgomery multiplication algorithm for ternary extension fields, GF (3 n), by essentially keeping the steps of the algorithm intact. Similarly, addition or inversion operations can be performed using similar algorithms that can be realized together in the same digital circuit. To summarize, an arithmetic module which is versatile in the sense that it can be adjusted to operate in more than one of the three fields is feasible, provided that this extra functionality does not lead to an excessive increase in area and dramatic decrease in speed. Quite contrarily, a unified module that is capable of performing arithmetic in more than one field in the same, unified datapath brings about many advantages, one of which is the improved {area × time} product.