The Impact of Applications on Mathematics by unknow

Author:unknow
Language: eng
Format: epub
Publisher: Springer Japan, Tokyo

5 Concluding Remarks

In this paper, we survey the recent progress of the discrete logarithm problem over small characteristic fields. The new index calculus methods [5, 17, 26] are much faster than the previous index calculus method like the function field sieve [3, 27] both in theory and in practice. The experiments and estimates [1, 2, 15, 16, 18–20, 24, 25] show the efficiency of these methods, and imply the pairing-based cryptosystems over small characteristic are totally insecure.

To keep the paring-based cryptosystems secure, one can use the pairings over large characteristic fields, such as Ate pairing [23]. In such case, the new index calculus cannot apply efficiently. However, the difficulty for solving the DLP over large characteristic extension fields related to pairing-based cryptosystems, such as GF for Barreto-Naehrig curves [8] and GF for Kachisa-Schaefer-Scott curves [28], is not precisely estimated yet, so that further research is needed.

Another important problem for paring-based cryptosystems is the pairing inversion problem, which is to compute the map of inverse of the pairing. The problem must also be intractable to make some cryptographic protocols secure. There are several papers [14, 34, 35] about the problem, however its difficulty is still not well-known.

In conclusion, secure use of the pairing-based cryptosystems requires further studies on the above mentioned points

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