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New discrete logarithm computation for the medium prime case using the function field sieve

  • * Corresponding author: Madhurima Mukhopadhyay

    * Corresponding author: Madhurima Mukhopadhyay 
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  • The present work reports progress in discrete logarithm computation for the general medium prime case using the function field sieve algorithm. A new record discrete logarithm computation over a 1051-bit field having a 22-bit characteristic was performed. This computation builds on and implements previously known techniques. Analysis indicates that the relation collection and descent steps are within reach for fields with 32-bit characteristic and moderate extension degrees. It is the linear algebra step which will dominate the computation time for any discrete logarithm computation over such fields.

    Mathematics Subject Classification: Primary: 11Y16; Secondary: 94A60.

    Citation:

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  • Table 1.  A comparison of the difficulty of computing discrete logarithms for the medium prime case using the function field sieve algorithm

    Ref $ \lceil\log_2 p\rceil $ $ n $ $ \lceil\log_2 p^n \rceil $ $ \lceil \log_2 \#\mathbb{B}\rceil $ $ \Lambda $
    JL [20] 17 25 401 18 3.79
    SS [24] 16 37 592 17 0.11
    SS [24] 19 40 728 20 0.08
    This work 22 50 1051 23 0.07
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    Table 2.  A comparison of the difficulty of computing discrete logarithms for the medium prime case using the function field sieve algorithm for Kummer extensions, i.e., for fields $\mathbb{F}_{p^n}$ satisfying $n\mathrel| (p-1)$

    Ref $\lceil\log_2 p\rceil$ $n$ $\lceil\log_2 p^n \rceil$ $\lceil \log_2 \#\mathbb{B}\rceil$ $\Lambda$
    JL[20]1930556184.29
    Joux[18]25471175200.77
    Joux[18]25571425200.13
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