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An algorithm for generalized syndrome decoding problem

  • *Corresponding author: Huimin Lao

    *Corresponding author: Huimin Lao 

This work was supported by the National Natural Science Foundation of China (No. 62032009), and the Guangdong Major Program of Basic and Applied Basic Research (No. 2019B030302008)

Abstract Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by
  • Syndrome decoding problem has received a lot of attention due to its applications in code-based cryptography. In this paper, we study a generalization of the syndrome decoding problem. This problem was recently introduced in Khathuria, Rosenthal, and Weger's cryptosystem. We present a new algorithm for the generalized syndrome decoding problem, which is adapted from the information set decoding algorithm proposed by Finiasz and Sendrier. Furthermore, our algorithm can be severed as an attack for Khathuria, Rosenthal, and Weger's cryptosystem. New parameters of their cryptosystem are suggested for which our algorithm needs $ 2^{128} $ and $ 2^{256} $ bit operations.

    Mathematics Subject Classification: Primary: 94A60, 94B35; Secondary: 14G50.

    Citation:

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  • Figure 1.  The complexity of our algorithm for different choices of parameters $ c_t $ and $ \lambda $

    Table 1.  The worse case complexity for $ q $ when $ \lambda = 2 $ and $ c_n = 0.5 $

    $ q $ $ 2 $ $ 7 $ $ 27 $ $ 49 $ $ 64 $ $ 81 $ $ 256 $
    complexity $ 0.05972 $ $ 0.06108 $ $ 0.06147 $ $ 0.06164 $ $ 0.06172 $ $ 0.06179 $ $ 0.06212 $
     | Show Table
    DownLoad: CSV

    Table 2.  The security of the KRW cryptosystem against our algorithm

    $ (q, n, r, t, \lambda) $ $ p $ $ s $ Complexity
    $ (7, 1872, 2920,103, 2) $ $ 4 $ $ 1468 $ $ 2^{251} $
    $ (13, 1258, 1835,114, 2) $ $ 4 $ $ 924 $ $ 2^{251} $
     | Show Table
    DownLoad: CSV

    Table 3.  New sets of parameters of the KRW cryptosystem that achieve 128-bit security level

    $ k/n $ $ q $ $ m $ $ \lambda $ $ n $ $ k $ Public key Size (Bits)
    0.75 7 4 2 829 622 1929367
    0.85 7 4 2 825 701 1599451
    0.75 7 7 3 1025 769 6454491
    0.85 7 7 3 907 771 4727832
    0.75 16 4 2 799 599 2553568
    0.85 16 4 2 802 682 2158080
     | Show Table
    DownLoad: CSV

    Table 4.  New sets of parameters of the KRW cryptosystem that} {achieve 256-bit security level

    $ k/n $ $ q $ $ m $ $ \lambda $ $ n $ $ k $ Public key Size (Bits)
    0.75 7 4 2 1764 1323 8735635
    0.85 7 4 2 1741 1480 7145482
    0.75 7 7 3 2246 1685 30989821
    0.85 7 7 3 1979 1682 22482084
    0.75 16 4 2 1750 1313 12249984
    0.85 16 4 2 1723 1465 9964992
     | Show Table
    DownLoad: CSV
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