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Hardware and Software Normal Basis Arithmetic for Pairing-Based Cryptography in Characteristic Three
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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
Efficient GF(pm) arithmetic architectures for cryptographic applications
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Reduction optimal trinomials for efficient software implementation of the ηT pairing
IWSEC'07 Proceedings of the Security 2nd international conference on Advances in information and computer security
Efficient tate pairing computation for elliptic curves over binary fields
ACISP'05 Proceedings of the 10th Australasian conference on Information Security and Privacy
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Designs, Codes and Cryptography
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The ηT pairing for supersingular elliptic curves over GF(3m) has been paid attention because of its computational efficiency. Since most computation parts of the ηT pairing are GF(3m) multiplications, it is important to improve the speed of the multiplication when implementing the ηT pairing. In this paper we investigate software implementation of GF(3m) multiplication and propose using irreducible trinomials xm + axk + b over GF(3) such that k is a multiple of w, where w is the bit length of the word of targeted CPU. We call the trinomials “reduction optimal trinomials (ROTs).” ROTs actually exist for several m's and for typical values of w = 16 and 32. We list them for extension degrees m = 97, 167, 193, 239, 317, and 487. These m's are derived from security considerations. Using ROTs, we are able to implement efficient modulo operations (reductions) for GF(3m) multiplication compared with cases in which other types of irreducible trinomials are used (e.g., trinomials with a minimum k for each m). The reason for this is that for cases using ROTs, the number of shift operations on multiple precision data is reduced to less than half compared with cases using other trinomials. Our implementation results show that programs of reduction specialized for ROTs are 20–30% faster on 32-bit CPU and approximately 40% faster on 16-bit CPU compared with programs using irreducible trinomials with general k.