Pairing based cryptography is a new public key cryptographic scheme. An elliptic curve suitable for pairing based cryptography is called a “pairing-friendly” elliptic curve. After Mitsunari, Sakai and Kasahara’s traitor tracing scheme and Boneh and Boyen’s short signature scheme, many protocols based on pairing-related problems such as the q-weak Diffie-Hellman problem have been proposed. In Eu...

A Nilsson mean-field plus extended pairing interaction Hamiltonian with many-pair interaction terms is proposed. Eigenvalues of the extended pairing model are easy to obtain. Our investigation shows that one- and two-body interactions continue to dominate the dynamics for relatively small values of the pairing strength. As the strength of the pairing interaction grows, however, the three- and h...

We propose an improved implementation of modified Weil pairings. By reduction of operations in the extension field to those in the base field, we can save some operations in the extension field when computing a modified Weil pairing. In particular, computing e`(P, φ(P )) is the same as computing the Tate pairing without the final powering. So we can save about 50% of time for computing e`(P, φ(...

This paper presents the first study of pairing computation on curves with embedding degree 15. We compute the Ate and the twisted Ate pairing for a family of curves with parameter ρ 1.5 and embedding degree 15. We use a twist of degree 3 to perform most of the operations in Fp or Fp5 . Furthermore, we present a new arithmetic for extension fields of degree 5. Our computations show that these cu...

We study superconductivity in spin-orbit-coupled systems in the vicinity of inversion symmetry breaking. We find that, because of the presence of spin-orbit coupling, fluctuations of the incipient parity-breaking order generate an attractive pairing interaction in an odd-parity pairing channel, which competes with the s-wave pairing. We show that Coulomb repulsion or an external Zeeman field su...

Let E be an elliptic curve over a finite field Fq with a power of prime q, r a prime dividing #E(Fq), and k the smallest positive integer satisfying r|Φk(p), called embedding degree. Then a bilinear map t : E(Fq)[r]×E(Fqk )/rE(Fqk )→ Fqk is defined, called the Tate pairing. And the Ate pairing and other variants are obtained by reducing the domain for each argument and raising it to some power....

The Tate pairing has plenty of attractive applications, e.g., ID-based cryptosystems, short signatures, etc. Recently several fast implementations of the Tate pairing has been reported, which make it appear that the Tate pairing is capable to be used in practical applications. The computation time of the Tate pairing strongly depends on underlying elliptic curves and definition fields. However ...

A significant amount of research has focused on methods to improve the efficiency of cryptographic pairings; in part this work is motivated by the wide range of applications for such primitives. Although numerous hardware accelerators for pairing evaluation have used parallelism within extension field arithmetic to improve efficiency, similar techniques have not been examined in software thus f...

The Weil and Tate pairings have been used recently to build new schemes in cryptography. It is known that the Weil pairing takes longer than twice the running time of the Tate pairing. Hence it is necessary to develop more efficient implementations of the Tate pairing for the practical application of pairing based cryptosystems. In 2002, Barreto et al. and Galbraith et al. provided new algorith...