TAILIEUCHUNG - Efficient Unified Arithmetic for Hardware Cryptography

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 (3n ). | Efficient Unified Arithmetic for Hardware Cryptography Erkay Savas1 and Jetin Kaya Koc2 1 Sabanci University erkays@ 2 Oregon State University koc@ The basic arithmetic operations . addition multiplication and inversion in finite fields GF q where q pk 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 2n . 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 3n . 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 2n given in 8 . In addition it is almost straightforward to extend the Montgomery multiplication algorithm for ternary extension fields GF 3n 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

• An ninh - Bảo mật
• Fundamentals of Extension Fields
• Addition and Subtraction
• Binary extension field
• Ternary extension field
• Montgomery Multiplication Algorithm
• Efficient Unified Arithmetic