The security of pairing-based cryptosystems is determined by the difficulty of solving the discrete logarithm problem (DLP) over certain types of finite fields. One of the most efficient algorithms for computing a pairing is the ηT pairing over supersingular curves on finite fields of characteristic 3. Indeed many high-speed implementations of this pairing have been reported, and it is an attractive candidate for practical deployment of pairing-based cryptosystems. Since the embedding degree of the ηT pairing is 6, we deal with the difficulty of solving a DLP over the finite field GF(36n), where the function field sieve (FFS) is known as the asymptotically fastest algorithm of solving it. Moreover, several efficient algorithms are employed for implementation of the FFS, such as the large prime variation. In this paper, we estimate the time complexity of solving the DLP for the extension degrees n=97, 163, 193, 239, 313, 353, and 509, when we use the improved FFS. To accomplish our aim, we present several new computable estimation formulas to compute the explicit number of special polynomials used in the improved FFS. Our estimation contributes to the evaluation for the key length of pairing-based cryptosystems using the ηT pairing.

Pairings on elliptic curves over finite fields are crucial for constructing various cryptographic schemes. The ηT pairing on supersingular curves over GF(3n) is particularly popular since it is efficiently implementable. Taking into account the Menezes-Okamoto-Vanstone attack, the discrete logarithm problem (DLP) in GF(36n) becomes a concern for the security of cryptosystems using ηT pairings in this case. In 2006, Joux and Lercier proposed a new variant of the function field sieve in the medium prime case, named JL06-FFS. We have, however, not yet found any practical implementations on JL06-FFS over GF(36n). Therefore, we first fulfill such an implementation and we successfully set a new record for solving the DLP in GF(36n), the DLP in GF(36·71) of 676-bit size. In addition, we also compare JL06-FFS and an earlier version, named JL02-FFS, with practical experiments. Our results confirm that the former is several times faster than the latter under certain conditions.

In this paper, we suggest that all pairings are in a group from an abstract angle. Based on the results, some new pairings with the short Miller loop are constructed for great efficiency. It is possible that our observation can be applied into other aspects of pairing-based cryptosystems.

This paper proposes new candidate one-way functions constructed with a certain type of endomorphisms on non-supersingular elliptic curves. We can show that the one-wayness of our proposed functions is equivalent to some special cases of the co-Diffie-Hellman assumption. Also a digital signature scheme is explicitly described using our proposed functions.

Non-supersingular elliptic curves are important for the security of pairing-based cryptosystems. But there are few suitable non-supersingular elliptic curves for pairing-based cryptosystems. This letter introduces a method which allows the existing method to generate more non-supersingular elliptic curves suitable for pairing-based cryptosystems when the embedding degree is 6.

This paper provides methods for construction of pairing-based cryptosystems based on non-supersingular elliptic curves.