Solving a DLP with Auxiliary Input with the ρ-Algorithm
The discrete logarithm problem with auxiliary input (DLPwAI) is a problem to find a positive integer α from elements G, αG, α d G in an additive cyclic group generated by G of prime order r and a positive integer d dividing r –1. In 2011, Sakemi et al. implemented Cheon’s algorithm for solving DLPwAI, and solved a DLPwAI in a group with 128-bit order r in about 131 hours with a single core on an elliptic curve defined over a prime finite field which is used in the TinyTate library for embedded cryptographic devices. However, since their implementation was based on Shanks’ Baby-step Giant-step (BSGS) algorithm as a sub-algorithm, it required a large amount of memory (246 GByte) so that it was concluded that applying other DLPwAIs with larger parameter is infeasible. In this paper, we implemented Cheon’s algorithm based on Pollard’s ρ-algorithm in order to reduce the required memory. As a result, we have succeeded solving the same DLPwAI in about 136 hours by a single core with less memory (0.5 MByte).
KeywordsGroup Operation Elliptic Curve Random Oracle Single Core Discrete Logarithm Problem
Unable to display preview. Download preview PDF.
- 6.Box, R., et al.: A Fast Easy Sort. Computer Journal of Byte Magazine 16(4), 315–320 (1991)Google Scholar
- 9.GNU MP, http://gmplib.org/
- 10.Izu, T., Takenaka, M., Yasuda, M.: Experimental Results on Cheon’s Algorithm. In: WAIS 2010, The Proceedings of ARES 2010, pp. 625–630. IEEE Computer Science (2010)Google Scholar
- 11.Izu, T., Takenaka, M., Yasuda, M.: Experimental Analysis of Cheon’s Algorithm against Pairing-Friendly Curves. In: AINA 2011, pp. 90–96. IEEE Computer Science (2011)Google Scholar
- 15.Oliveira, L., López, J., Dahab, R.: TinyTate: Identity-Based Encryption for Sensor Networks. IACR Cryptology ePrint Archive, Report 2007/020 (2007)Google Scholar
- 18.Shanks, D.: Class Number, a Theory of Factorization, and Genera. In: Proc. of Symp. Math. Soc., vol. 20, pp. 41–440 (1971)Google Scholar