Complexity of Lattice Problems
Complexity of Lattice Problems
Evidence that XTR Is More Secure than Supersingular Elliptic Curve Cryptosystems
Journal of Cryptology
The Weil Pairing, and Its Efficient Calculation
Journal of Cryptology
Elliptic Curves: Number Theory and Cryptography, Second Edition
Elliptic Curves: Number Theory and Cryptography, Second Edition
Pairing '08 Proceedings of the 2nd international conference on Pairing-Based Cryptography
A Taxonomy of Pairing-Friendly Elliptic Curves
Journal of Cryptology
IEEE Transactions on Information Theory
Computing bilinear pairings on elliptic curves with automorphisms
Designs, Codes and Cryptography
A variant of Miller's formula and algorithm
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
Hidden pairings and trapdoor DDH groups
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Evaluating 2-DNF formulas on ciphertexts
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
Converting pairing-based cryptosystems from composite-order groups to prime-order groups
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
Pairing-Based cryptography at high security levels
IMA'05 Proceedings of the 10th international conference on Cryptography and Coding
Pairings on elliptic curves over finite commutative rings
IMA'05 Proceedings of the 10th international conference on Cryptography and Coding
IEEE Transactions on Information Theory
On efficient pairings on elliptic curves over extension fields
Pairing'12 Proceedings of the 5th international conference on Pairing-Based Cryptography
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In pairing-based cryptography, most researches are focused on elliptic curves of embedding degrees greater than six, but less on curves of small embedding degrees, although they are important for pairing-based cryptography over composite-order groups. This paper analyzes efficient pairings on ordinary elliptic curves of embedding degree 1 and 2 from the point of shortening Miller's loop. We first show that pairing lattices presented by Hess can be redefined on composite-order groups. Then we give a simpler variant of the Weil pairing lattice which can also be regarded as an Omega pairing lattice, and extend it to ordinary curves of embedding degree 1. In our analysis, the optimal Omega pairing, as the super-optimal pairing on elliptic curves of embedding degree 1 and 2, could be more efficient than Weil and Tate pairings. On the other hand, elliptic curves of embedding degree 2 are also very useful for pairings on elliptic curves over RSA rings proposed by Galbraith and McKee. So we analyze the construction of such curves over RSA rings, and redefine pairing lattices over RSA rings. Specially, modified Omega pairing lattices over RSA rings can be computed without knowing the RSA trapdoor. Furthermore, for keeping the trapdoor secret, we develop an original idea of evaluating pairings without leaking the group order.