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On the hardness of the Lee syndrome decoding problem

  • * Corresponding author: Violetta Weger

    * Corresponding author: Violetta Weger 
Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
  • In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric. We first prove that the decisional version of the problem is NP-complete, by a reduction from the $ 3 $-dimensional matching problem. Then, we study the complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite fields to the Lee metric over finite rings, as well as proposing some novel solutions. For the analyzed algorithms, we assess the computational complexity in the asymptotic regime and compare it to the corresponding algorithms in the Hamming metric.

    Mathematics Subject Classification: 11T71, 94B35.

    Citation:

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  • Figure 1.  Illustration of the error vectors at each level of the representation technique algorithm for two levels. At each level, the striped region represents the overlapping part

    Figure 2.  Comparing asymptotic complexity of different algorithms for different values of $ \lambda $. The values are calculated for $ q = 7^2 $

    Table 1.  Comparison of the asymptotic complexity of all the algorithms at rate $ R^* = {\rm{argmax}} _{0 \leq R \leq 1} \left( e(R,q) \right) $. The values are calculated for $ q = 7^2 $

    $ \lambda = 1 $ $ \lambda = 0.75 $ $ \lambda = 0.5 $
    $ R^* $ $ e(R^*,q) $ $ R^* $ $ e(R^*,q) $ $ R^* $ $ e(R^*,q) $
    Two-Blocks 0.3886 0.0913 0.4473 0.0978 0.4694 0.1211
    $ s $-Blocks 0.3969 0.1030 0.3441 0.0745 0.3441 0.07453
    Wagner $ a=1 $ 0.3925 0.0897 0.4473 0.0978 0.4694 0.1211
    Wagner $ a=2 $ 0.3925 0.0897 0.4473 0.0978 0.4694 0.1211
    Rep. tech. $ a=1 $ 0.3896 0.0998 0.4288 0.1155 0.4648 0.1457
    Rep. tech. $ a=2 $ 0.3922 0.1012 0.4275 0.1221 0.4757 0.1557
    BJMM level 2 0.4414 0.07440 0.4587 0.0954 0.4801 0.1178
    BJMM level 3 0.3921 0.1012 0.4282 0.1220 0.4754 0.1554
     | Show Table
    DownLoad: CSV

    Table 2.  Comparison with Hamming metric for $ q = 4 $ and $ \lambda = 1 $. The values for Hamming metric ISD algorithms BJMM-MO and Stern are from [20,Table 3] and [21,Table 1], respectively

    $ e(R^*,q) $
    Lee Metric
    Prange 0.0575
    $ s $-Blocks 0.0575
    Two-Blocks 0.0556
    Wagner $ a=1 $ 0.0556
    Rep. tech. $ a=1 $ 0.0569
    Rep. tech. $ a=2 $ 0.0571
    BJMM level 2 0.05265
    BJMM level 3 0.0557
    Hamming Metric
    BJMM-MO 0.04294
    Stern 0.04987
    Prange 0.05095
     | Show Table
    DownLoad: CSV
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