**Key Length Estimation of Pairing-based Cryptosystems using $\eta_T$ Pairing**

*Naoyuki Shinohara and Takeshi Shimoyama and Takuya Hayashi and Tsuyoshi Takagi *

**Abstract: **The security of pairing-based cryptosystems depends on the difficulty of the discrete logarithm problem (DLP) over certain types of finite fields. One of the most efficient algorithms for computing a pairing is the $\eta_T$ pairing over supersingular curves on finite fields whose characteristic is $3$. Indeed many high-speed implementations of this pairing have been reported, and it is an attractive candidate for practical deployment of pairing-based cryptosystems. The embedding degree of the $\eta_T$ pairing is 6, so we deal with the difficulty of a DLP over the finite field $ GF(3^{6n})$, 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 $n=97,163, 193,239,313,353,509$,
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 $\eta_T$ pairing.

**Category / Keywords: **public-key cryptography / pairing-based cryptosystems, discrete logarithm problem, finite field, key length, suitable values

**Publication Info: **This is a full version of ISPEC 2012 paper.

**Date: **received 25 Jan 2012, last revised 18 Jun 2012

**Contact author: **shnhr at nict go jp

**Available format(s): **PDF | BibTeX Citation

**Note: **Table 1 and 3 are edited.

**Version: **20120619:054704 (All versions of this report)

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