**Factor Base Discrete Logarithms in Kummer Extensions**

*Dianyan Xiao and Jincheng Zhuang and Qi Cheng*

**Abstract: **The discrete logarithm over finite fields of small characteristic can be solved much more efficiently than previously thought. This algorithmic breakthrough is based on pinpointing relations among the factor base discrete logarithms. In this paper, we concentrate on the Kummer extension $ \F_{q^{2(q-1)}}=\F_{q^2}[x]/(x^{q-1}-A). $ It has been suggested that in this case, a small number of degenerate relations (from the Borel subgroup) are enough to solve the factor base discrete logarithms. We disprove the conjecture, and design a new heuristic algorithm with an improved bit complexity $ \tilde{O}(q^{1+ \theta} ) $ (or algebraic complexity $\tilde{O}(q^{\theta} )$) to compute discrete logarithms of all the elements in the factor base $\{ x+\alpha | \alpha \in \F_{q^2} \} $, where $ \theta<2.38 $ is the matrix multiplication exponent over rings. Given additional time $ \tilde{O} (q^4), $
we can compute discrete logarithms of at least $ \Omega(q^3) $ many monic irreducible quadratic polynomials. We reduce the correctness of the algorithm to a conjecture concerning the determinant of a simple $ (q+1)$-dimensional lattice, rather than to elusive smoothness assumptions. We verify the conjecture numerically for all prime powers $ q $ such that $ \log_2(q^{2(q-1)}) \leq 5134 $, and provide theoretical
supporting evidences.

**Category / Keywords: **Discrete logarithms, Finite fields, Kummer extension, Character Sum

**Date: **received 5 Sep 2015, last revised 26 Feb 2017

**Contact author: **zhuangjincheng at iie ac cn

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

**Note: **19 pages, writing revised, appendix modified

**Version: **20170227:065232 (All versions of this report)

**Short URL: **ia.cr/2015/859

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