For ID-based cryptography, not only pairing but also scalar multiplication must be efficiently computable. In this paper, we propose a scalar multiplication method on the circumstances that we work at Ate pairing with Barreto-Naehrig (BN) curve. Note that the parameters of BN curve are given by a certain integer, namely mother parameter. Adhering the authors' previous policy that we execute scalar multiplication on subfield-twisted curve (Fp2) instead of doing on the original curve E(Fp12), we at first show sextic twisted subfield Frobenius mapping (ST-SFM) in (Fp2). On BN curves, note is identified with the scalar multiplication by p. However a scalar is always smaller than the order r of BN curve for Ate pairing, so ST-SFM does not directly applicable to the above circumstances. We then exploit the expressions of the curve order r and the characteristic p by the mother parameter to derive some radices such that they are expressed as a polynomial of p. Thus, a scalar multiplication [s] can be written by the series of ST-SFMs . In combination with the binary method or multi-exponentiation technique, this paper shows that the proposed method runs about twice or more faster than plain binary method.

It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log 2r/(k), where () is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from ⌊log 2r⌋ to ⌊ log 2(t-1)⌋, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "χ." For such a curve, this paper gives integer variable χ-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve, it reduces the number of loops to ⌊log 2χ⌋. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.

EMV signature is one of specifications for authenticating credit and debit card data, which is based on ISO/IEC 9796-2 signature scheme. At CRYPTO 2009, Coron, Naccache, Tibouchi, and Weinmann proposed a new forgery attack against the signature ISO/IEC 9796-2 (CNTW attack) [2]. They also briefly discussed the possibility when the attack is applied to the EMV signatures. They showed that the forging cost is $45,000 and concluded that the attack could not forge them for operational reason. However their results are derived from not fully analysis under only one condition. The condition they adopt is typical case. For security evaluation, fully analysis and an estimation in worst case are needed. This paper shows cost-estimation of CNTW attack against EMV signature in detail. We constitute an evaluate model and show cost-estimations under all conditions that Coron et al. do not estimate. As results, this paper contribute on two points. One is that our detailed estimation reduced the forgery cost from $45,000 to $35,200 with same condition as [2]. Another is to clarify a fact that EMV signature can be forged with less than $2,000 according to a condition. This fact shows that CNTW attack might be a realistic threat.

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) ⌊log2 r⌋. On the other hand, the twisted Ate pairing requires (3/4) ⌊log2 r⌋ 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 proposed idea 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) ⌊log2 r ⌋ attained by the most efficient Ate pairings.