Efficient Algorithms for Pairing-Based Cryptosystems
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
An Elliptic Curve Implementation of the Finite Field Digital Signature Algorithm
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
A One Round Protocol for Tripartite Diffie-Hellman
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Hardware Implementation of Finite Fields of Characteristic Three
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
Irreducible trinomials over finite fields
Mathematics of Computation
Hardware and Software Normal Basis Arithmetic for Pairing-Based Cryptography in Characteristic Three
IEEE Transactions on Computers
Efficient pairing computation on supersingular Abelian varieties
Designs, Codes and Cryptography
An Algorithm for the nt Pairing Calculation in Characteristic Three and its Hardware Implementation
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
Some efficient algorithms for the final exponentiation of ηT pairing
ISPEC'07 Proceedings of the 3rd international conference on Information security practice and experience
Efficient GF(pm) arithmetic architectures for cryptographic applications
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Efficient tate pairing computation for elliptic curves over binary fields
ACISP'05 Proceedings of the 10th Australasian conference on Information Security and Privacy
Collusion resistant broadcast encryption with short ciphertexts and private keys
CRYPTO'05 Proceedings of the 25th annual international conference on Advances in Cryptology
Reduction Optimal Trinomials for Efficient Software Implementation of the ηT Pairing
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
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The ηT pairing for supersingular elliptic curve over GF(3m) has been paid attention because of its computational efficiency. Since most parts of computation of the ηT pairing are multiplications over GF(3m), it is important to improve the speed of the multiplication when implementing the ηT pairing. In this paper we consider software implementation of multiplication over GF(3m) and propose to use irreducible trinomials xm + axk + b over GF(3) such that w, bit length of word of targeted CPU, divides k. We call the trinomials "reduction optimal trinomials (ROTs)". ROTs actually exist for several m's and typical values of w = 16 and 32. We list them for extension degrees m = 97, 167, 193 and 239. These m's are derived from security considerations. Using ROT it is possible to implement efficient modulo operation (reduction) in multiplication over GF(3m) comparing with the case using other type of trinomials (e.g., trinomials with minimum k for each m). The reason of this is that for the cases of reduction by ROT the number of shift operations on multiple precision data reduces to less than half comparing with the cases by other trinomials. Implementation results show that reduction algorithm specialized for ROT is 20-30% faster on 32-bit CPU and around 40% faster on 16-bit CPU than algorithm for irreducible trinomials with general k.