## Cryptology ePrint Archive: Report 2003/019

A Polynomial Time Algorithm for the Braid Diffie-Hellman Conjugacy Problem

Jung Hee Cheon and Byungheup Jun

Abstract: We propose the first polynomial time algorithm for the braid Diffie-Hellman conjugacy problem (DHCP) on which the braid key exchange scheme and the braid encryption scheme are based~\cite{KLCHKP01}. We show the proposed method solves the DHCP for the image of braids under the Lawrence-Krammer representation and the solutions play the equivalent role of the original key for the DHCP of braids. Given a braid index $n$ and a canonical length $\ell$, the complexity is about $2^{-2}\ell^3 n^{4\tau+2}\log n$ bit operations, where $\tau=\log_27\approx 2.8$ (Theoretically, it can be reduced to $O(\ell^3 n^{8.3}\log n)$ using $\tau=2.376$). Further, we show that the generalization into the decomposition problem causes only 8 times of the complexity.

Category / Keywords: public-key cryptography / cryptanalysis

Date: received 28 Jan 2003, last revised 8 Apr 2003

Contact author: jhcheon at icu ac kr

Available format(s): Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

Note: We improved the complexity of the algorithm and generalized to the decomposition problem.

Short URL: ia.cr/2003/019

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