Paper 2012/345
Breaking pairing-based cryptosystems using $\eta_T$ pairing over $GF(3^{97})$
Takuya Hayashi, Takeshi Shimoyama, Naoyuki Shinohara, and Tsuyoshi Takagi
Abstract
There are many useful cryptographic schemes, such as ID-based encryption, short signature, keyword searchable encryption, attribute-based encryption, functional encryption, that use a bilinear pairing. It is important to estimate the security of such pairing-based cryptosystems in cryptography. The most essential number-theoretic problem in pairing-based cryptosystems is the discrete logarithm problem (DLP) because pairing-based cryptosystems are no longer secure once the underlining DLP is broken. One efficient bilinear pairing is the $\eta_T$ pairing defined over a supersingular elliptic curve $E$ on the finite field $GF(3^n)$ for a positive integer $n$. The embedding degree of the $\eta_T$ pairing is $6$; thus, we can reduce the DLP over $E$ on $GF(3^n)$ to that over the finite field $GF(3^{6n})$. In this paper, for breaking the $\eta_T$ pairing over $GF(3^n)$, we discuss solving the DLP over $GF(3^{6n})$ by using the function field sieve (FFS), which is the asymptotically fastest algorithm for solving a DLP over finite fields of small characteristics. We chose the extension degree $n=97$ because it has been intensively used in benchmarking tests for the implementation of the $\eta_T$ pairing, and the order (923-bit) of $GF(3^{6\cdot 97})$ is substantially larger than the previous world record (676-bit) of solving the DLP by using the FFS. We implemented the FFS for the medium prime case (JL06-FFS), and propose several improvements of the FFS, for example, the lattice sieve for JL06-FFS and the filtering adjusted to the Galois action. Finally, we succeeded in solving the DLP over $GF(3^{6\cdot 97})$. The entire computational time of our improved FFS requires about 148.2 days using 252 CPU cores. Our computational results contribute to the secure use of pairing-based cryptosystems with the $\eta_T$ pairing.
Metadata
- Available format(s)
-
PDF
- Category
- Public-key cryptography
- Publication info
- Published elsewhere. Unknown where it was published
- Keywords
- pairing-based cryptosystems$\eta_T$ pairingdiscrete logarithm problemsfunction filed sieve
- Contact author(s)
- t-hayashi @ math kyushu-u ac jp
- History
- 2012-06-22: received
- Short URL
- https://ia.cr/2012/345
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2012/345, author = {Takuya Hayashi and Takeshi Shimoyama and Naoyuki Shinohara and Tsuyoshi Takagi}, title = {Breaking pairing-based cryptosystems using $\e{ta_T}$ pairing over ${GF}(3^{97})$}, howpublished = {Cryptology {ePrint} Archive, Paper 2012/345}, year = {2012}, url = {https://eprint.iacr.org/2012/345} }