Key Length Estimation of Pairing-Based Cryptosystems Using <i>η<sub>T</sub></i> Pairing over <i>GF</i>(3<i><sup>n</sup></i>)

Abstract

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.