A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases
Information and Computation
A course in computational algebraic number theory
A course in computational algebraic number theory
Short Signatures from the Weil Pairing
ASIACRYPT '01 Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Integer Variable Χ---Based Ate Pairing
Pairing '08 Proceedings of the 2nd international conference on Pairing-Based Cryptography
Exponentiation in Pairing-Friendly Groups Using Homomorphisms
Pairing '08 Proceedings of the 2nd international conference on Pairing-Based Cryptography
Skew Frobenius Map and Efficient Scalar Multiplication for Pairing---Based Cryptography
CANS '08 Proceedings of the 7th International Conference on Cryptology and Network Security
On Software Parallel Implementation of Cryptographic Pairings
Selected Areas in Cryptography
Efficient and generalized pairing computation on Abelian varieties
IEEE Transactions on Information Theory
Optimised versions of the ate and twisted ate pairings
Cryptography and Coding'07 Proceedings of the 11th IMA international conference on Cryptography and coding
Verifier-Local revocation group signature schemes with backward unlinkability from bilinear maps
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
Constructing pairing-friendly elliptic curves with embedding degree 10
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Hierarchical identity based encryption with constant size ciphertext
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Pairing-Friendly elliptic curves of prime order
SAC'05 Proceedings of the 12th international conference on Selected Areas in Cryptography
IEEE Transactions on Information Theory
Implementing cryptographic pairings over barreto-naehrig curves
Pairing'07 Proceedings of the First international conference on Pairing-Based Cryptography
On efficient pairings on elliptic curves over extension fields
Pairing'12 Proceedings of the 5th international conference on Pairing-Based Cryptography
An improved twisted ate pairing over KSS curves with k=18
Pairing'12 Proceedings of the 5th international conference on Pairing-Based Cryptography
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In the case of Barreto-Naehrig pairing-friendly curves of embedding degree 12 of order r, recent efficient Ate pairings such as R-ate, optimal, and Xate pairings achieve Miller loop lengths of (1/4) ⌊log2r⌋. On the other hand, the twisted Ate pairing requires (3/4) ⌊log2r⌋ loop iterations, and thus is usually slower than the recent efficient Ate pairings. This paper proposes an improved twisted Ate pairing using Frobenius maps and a small scalar multiplication. The proposal splits the Miller's algorithm calculation into several independent parts, for which multi-pairing techniques apply efficiently. The maximum number of loop iterations in Miller's algorithm for the proposed twisted Ate pairing is equal to the (1/4) ⌊log2r⌋ attained by the most efficient Ate pairings.