\`x^2+y_1+z_12^34\`
Advanced Search
Advanced Search
Advanced Search
This issue Previous Article Next Article
Article Contents
Article Contents
2022, Volume 16, Issue 4: 1023-1049. Doi: 10.3934/amc.2022094
This issue Previous Article Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes Next Article Optimal binary linear codes from posets of the disjoint union of two chains

On construction of lightweight MDS matrices

  • 1.

    Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China

  • *Corresponding author: Yuan Chen

    *Corresponding author: Yuan Chen 
Received:  March 2022
Revised:  November 2022
Early access:  November 2022
Published:  November 2022

The fourth author is supported by [Application Foundation Frontier Project of Wuhan Science and Technology Bureau under Grant 2020010601012189 and the National Natural Science Foundation of China under Grant 62072161]

Abstract / Introduction Full Text(HTML) Figure(2) / Table(8) Related Papers Cited by
    Abstract

  • MDS matrices are widely used in block ciphers. Constructing lightweight MDS matrices is one of the research focuses of lightweight cryptography. In this paper, we define a new operation called the Copy operation by using registers. It is a generalization of Type 3 elementary operations (add a row to another one multiplied by a nonzero number). It is shown that any nonsingular matrix can be obtained by Copy operations and Multiplication operations from the identity matrix $ I $ (a Copy Block Implementation of the matrix). Thus we introduce a new metric called gw-xor using Copy Block Implementations to construct lightweight MDS matrices with respect to low xor gates. Compared with sw-xor, the gw-xor count is a better approximation of the optimal implementation cost, and in particular it may be a better approximation of the optimal implementation cost than s-xor. By searching the potential paths of Copy operations that can obtain formal MDS matrices (i.e., matrices with indeterminate elements and each determinant of square submatrix of any order is a nonzero polynomial in these indeterminates), we find 52 classes $ 16\times 16 $ and $ 32\times 32 $ binary MDS matrices with 35 and 67 xor gates respectively, which are the best known results. Furthermore, by considering the depth of MDS matrices, we find more $ 4\times4 $ MDS matrices over $ \mathbb{F}_{2^n} $ with the lowest xor gates at depths 3, 4, 5.

    Mathematics Subject Classification: Primary: 11T71, 68P25, 94B60.

    Citation:
    shu

    \begin{equation} \\ \end{equation}
    • Full Text(HTML)
  • 加载中

  • Figure 1.  The implementation of the path $ \mathfrak{P}_1 $ in Example 3

    Download: Full-size image PowerPoint slide

    Figure 2.  The circuit implementation in Example 4

    Download: Full-size image PowerPoint slide

    Table 1.  An implementation of the matrix $ M $ in Example 2

    1 $ x_5\oplus x_9\rightarrow x_{17} $ 2 $ x_6\oplus x_{10}\rightarrow x_{18}(y_7) $ 3 $ x_{7}\oplus x_{11}\rightarrow x_{19}(y_8) $
    4 $ x_{8}\oplus x_{12}\rightarrow x_{20}(y_5) $ 5 $ x_{13}\oplus x_{16}\rightarrow x_{21} $ 6 $ x_{16}\oplus x_{9}\rightarrow x_{22}(y_9) $
    7 $ x_{21}\oplus x_{10}\rightarrow x_{23}(y_{10}) $ 8 $ x_{11}\oplus x_{14}\rightarrow x_{24}(y_{11}) $ 9 $ x_{12}\oplus x_{15}\rightarrow x_{25}(y_{12}) $
    10 $ x_{17}\oplus x_{20}\rightarrow x_{26}(y_6) $ 11 $ x_{9}\oplus x_{12}\rightarrow x_{27} $ 12 $ x_{13}\oplus x_{12}\rightarrow x_{28}(y_{13}) $
    13 $ x_{14}\oplus x_{27}\rightarrow x_{29}(y_{14}) $ 14 $ x_{15}\oplus x_{10}\rightarrow x_{30}(y_{15}) $ 15 $ x_{11}\oplus x_{16}\rightarrow x_{31}(y_{16}) $
     | Show Table
    DownLoad: CSV

    Table 2.  Comparison of metrics, where $ \alpha $ is a root of the polynomial $ x^4+x+1 $

    Matrix over $ \mathbb{F}_2[x]/x^4+x+1 $ sw-xor s-xor gw-xor g-xor Reference
    $ \left( {\begin{array}{*{20}{c}} \boldsymbol{1}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{\alpha}&\boldsymbol{\alpha}&\boldsymbol{0}\\ \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{1}&\boldsymbol{\alpha}\\ \boldsymbol{0}&\boldsymbol{0}&\boldsymbol{\alpha}&\boldsymbol{1} \end{array}} \right) $ 16 16 15 15 Example 2
    $ \left( {\begin{array}{*{20}{c}} \boldsymbol{1}&\boldsymbol{1}&\boldsymbol{1}&\boldsymbol{\alpha^3+1}\\ \boldsymbol{\alpha}&\boldsymbol{a+1}&\boldsymbol{1}&\boldsymbol{\alpha^3+\alpha+1}\\ \boldsymbol{\alpha^3+\alpha}&\boldsymbol{a+1}&\boldsymbol{\alpha^3+\alpha^2+1}&\boldsymbol{\alpha^3+\alpha^2}\\ \boldsymbol{\alpha+1}&\boldsymbol{\alpha}&\boldsymbol{\alpha^3+1}&\boldsymbol{\alpha^3+\alpha^2+\alpha+1} \end{array}} \right) $ 36 [24] $ \leq 35 $ [26] 36 $ \leq35 $ [26]
     | Show Table
    DownLoad: CSV

    Table 3.  The potential paths for $ 4\times4 $ MDS matrices

    No Representative path No Representative path
    1 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ 2 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_2} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{11}} \right) $
    3 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{10}} \right) $ 4 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{11}} \right) $
    5 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4,{10}} \right) $ 6 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{1, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4,{10}} \right) $
    7 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,{5}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ 8 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{10}} \right) $
    9 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ 10 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{1}} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{1, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $
    11 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{1}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ 12 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_7} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4,{10}} \right) $
    13 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ 14 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_8} , \mathfrak{R}_{1, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{11}} \right) $
    15 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{1, \overline{\mathbf{r}}_7} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ 16 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{1}} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{11}} \right) $
    17 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8}, \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ 18 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{11}} \right) $
     | Show Table
    DownLoad: CSV

    Table 4.  The 52 classes of MDS matrices with $ 8n+3 $ xor gates

    No. Path $ T $ $ \mathbf{U} $
    1 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A , \alpha_{32}= A , \beta_{5}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    2 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A^{-1} , \alpha_{32}= A^{-1} , \beta_{5}= A $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    3 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A^{-1} , \alpha_{41}= A , \beta_{7}= A $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    4 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A , \alpha_{41}= A^{-1} , \beta_{7}= A^{-1} $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    5 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{32}= A^{-1} , \beta_{3}= A , \alpha_{64}= A $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    6 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{32}= A , \beta_{3}= A^{-1} , \alpha_{64}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    7 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{41}= A , \beta_{4}= A , \alpha_{64}= A $ $ \{0, 0, 0, 0, 1, 0, 1, 0\} $
    8 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{3, \overline{\mathbf{r}}_6} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{41}= A^{-1} , \beta_{4}= A^{-1} , \alpha_{64}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    9 [10,24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A^{-1} , \beta_{4}= A , \beta_{5}= A $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    10 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A , \beta_{4}= A^{-1} , \beta_{5}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    11 [10,24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{10}} \right) $ $ \beta_{3}= A , \alpha_{54}= A , \beta_{5}= A $ $ \{0, 0, 0, 0, 0, 1, 0, 0\} $
    12 [24] $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{10}} \right) $ $ \beta_{3}= A^{-1} , \alpha_{54}= A^{-1} , \beta_{5}= A^{-1} $ $ \{0, 0, 0, 0, 0, 1, 0, 0\} $
    13 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A^{-1} , \alpha_{52}= A , \alpha_{64}= A $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    14 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A , \alpha_{52}= A^{-1} , \alpha_{64}= A^{-1} $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    15 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A , \beta_{5}= A^{-1} , \beta_{6}= A $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    16 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A^{-1} , \beta_{5}= A , \beta_{6}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    17 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{33}= A^{-1} , \alpha_{52}= A , \alpha_{64}= A $ $ \{0, 0, 0, 0, 1, 1, 1, 0\} $
    18 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{33}= A , \alpha_{52}= A^{-1} , \alpha_{64}= A^{-1} $ $ \{0, 0, 0, 0, 1, 1, 1, 0\} $
    19 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \beta_{4}= A , \alpha_{64}= A , \beta_{6}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    20 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{2,\overline{\mathbf{r_{5}}}}, \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{2, \overline{\mathbf{r}}_{10}} \right) $ $ \beta_{4}= A^{-1} , \alpha_{64}= A^{-1} , \beta_{6}= A $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    21 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A , \alpha_{54}= A , \alpha_{62}= A $ $ \{0, 0, 0, 0, 1, 0, 1, 0\} $
    22 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{23}= A^{-1} , \alpha_{54}= A^{-1} , \alpha_{62}= A^{-1} $ $ \{0, 0, 0, 0, 1, 0, 1, 0\} $
    23 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{33}= A , \alpha_{54}= A , \alpha_{62}= A $ $ \{0, 0, 0, 1, 1, 0, 0, 0\} $
    24 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{33}= A^{-1} , \alpha_{54}= A^{-1} , \alpha_{62}= A^{-1} $ $ \{0, 0, 0, 1, 1, 0, 0, 0\} $
    25 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{33}= A , \beta_{5}= A , \beta_{6}= A^{-1} $ $ \{0, 0, 0, 1, 0, 0, 0, 0\} $
    26 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \alpha_{33}= A^{-1} , \beta_{5}= A^{-1} , \beta_{6}= A $ $ \{0, 0, 0, 1, 0, 0, 0, 0\} $
    27 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \beta_{4}= A , \alpha_{62}= A , \beta_{6}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    28 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{3, \overline{\mathbf{r}}_2} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_8} , \mathfrak{R}_{2, \overline{\mathbf{r}}_5} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{4, \overline{\mathbf{r}}_{10}} \right) $ $ \beta_{4}= A^{-1} , \alpha_{62}= A^{-1} , \beta_{6}= A $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    29 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A^{-1} , \beta_{3}= A , \beta_{5}= A $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    30 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A , \beta_{3}= A^{-1} , \beta_{5}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    31 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A , \beta_{3}= A , \alpha_{64}= A $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    32 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A^{-1} , \beta_{3}= A^{-1} , \alpha_{64}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    33 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A , \alpha_{42}= A , \beta_{7}= A $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    34 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A^{-1} , \alpha_{42}= A^{-1} , \beta_{7}= A^{-1} $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    35 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{31}= A , \beta_{5}= A , \beta_{7}= A $ $ \{0, 0, 1, 0, 0, 0, 0, 0\} $
    36 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_2} , \mathfrak{R}_{3, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{31}= A^{-1} , \beta_{5}= A^{-1} , \beta_{7}= A^{-1} $ $ \{0, 0, 1, 0, 0, 0, 0, 0\} $
    37 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A , \alpha_{31}= A^{-1} , \alpha_{64}= A $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    38 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A^{-1} , \alpha_{31}= A , \alpha_{64}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 1, 0\} $
    39 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A , \alpha_{31}= A , \beta_{5}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    40 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A^{-1} , \alpha_{31}= A^{-1} , \beta_{5}= A $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    41 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A^{-1} , \alpha_{53}= A , \beta_{7}= A $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    42 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A , \alpha_{53}= A^{-1} , \beta_{7}= A^{-1} $ $ \{0, 0, 0, 0, 1, 0, 0, 0\} $
    43 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{31}= A^{-1} , \alpha_{53}= A , \alpha_{64}= A $ $ \{0, 0, 1, 0, 1, 0, 1, 0\} $
    44 $ \left(\mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{2, \overline{\mathbf{r}}_6} , \mathfrak{R}_{3, \overline{\mathbf{r}}_7} , \mathfrak{R}_{4, \overline{\mathbf{r}}_8} , \mathfrak{R}_{4,\overline{\mathbf{r_{1}}}} , \mathfrak{R}_{1, \overline{\mathbf{r}}_9} , \mathfrak{R}_{3, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{31}= A , \alpha_{53}= A^{-1} , \alpha_{64}= A^{-1} $ $ \{0, 0, 1, 0, 1, 0, 1, 0\} $
    45 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A^{-1} , \alpha_{31}= A , \alpha_{44}= A $ $ \{0, 0, 1, 0, 0, 0, 0, 0\} $
    46 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A , \alpha_{31}= A^{-1} , \alpha_{44}= A^{-1} $ $ \{0, 0, 1, 0, 0, 0, 0, 0\} $
    47 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A , \beta_{3}= A^{-1} , \beta_{7}= A $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    48 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{23}= A^{-1} , \beta_{3}= A , \beta_{7}= A^{-1} $ $ \{0, 0, 0, 0, 0, 0, 0, 0\} $
    49 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A , \alpha_{44}= A , \beta_{7}= A^{-1} $ $ \{0, 0, 0, 1, 0, 0, 0, 0\} $
    50 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \beta_{2}= A^{-1} , \alpha_{44}= A^{-1} , \beta_{7}= A $ $ \{0, 0, 0, 1, 0, 0, 0, 0\} $
    51 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{31}= A , \alpha_{44}= A , \alpha_{63}= A^{-1} $ $ \{0, 0, 1, 1, 0, 0, 1, 0\} $
    52 $ \left( \mathfrak{R}_{3, \overline{\mathbf{r}}_4} , \mathfrak{R}_{1, \overline{\mathbf{r}}_5} , \mathfrak{R}_{4, \overline{\mathbf{r}}_6} , \mathfrak{R}_{2, \overline{\mathbf{r}}_7} , \mathfrak{R}_{3, \overline{\mathbf{r}}_8} , \mathfrak{R}_{3, \overline{\mathbf{r}}_1} , \mathfrak{R}_{4, \overline{\mathbf{r}}_9} , \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}} \right) $ $ \alpha_{31}= A^{-1} , \alpha_{44}= A^{-1} , \alpha_{63}= A $ $ \{0, 0, 1, 1, 0, 0, 1, 0\} $
    Here $A$ is the companion matrix of the minimal polynomial $x^4+x+1$ or $x^8+x^2+1$.
     | Show Table
    DownLoad: CSV

    Table 5.  The depth calculation of $ M = (\mathfrak{P}_1,\mathbf{S},\mathbf{U}) $ in Example 4

    Copy operation Transformation Depth Copy operation Transformation Depth
    $ D_1=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1}] $ - [0, 0, 0, 0] $ D_5=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1}] $ - [3, 1, 2, 0]
    $ \mathfrak{R}_{3, \overline{\mathbf{r}}_4}(\mathbf{1}) $ $ x_3(\overline{\mathbf{r}}_5)\leftarrow x_3+\overline{\mathbf{r}}_4 $ [0, 0, 1, 0] $ \mathfrak{R}_{4, \overline{\mathbf{r}}_7}(\mathbf{1}) $ $ x_4(\overline{\mathbf{r}}_9)\leftarrow x_4+\overline{\mathbf{r}}_7 $ [3, 1, 2, 4]
    $ D_2=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1}] $ - [0, 0, 1, 0] $ D_6=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{ \pmb{\mathsf{ α}}} ] $ $ x_4(\overline{\mathbf{r}}_{9}^1)\leftarrow \mathbf{ \pmb{\mathsf{ α}}} x_4 $ [3, 1, 2, 5]
    $ \mathfrak{R}_{2, \overline{\mathbf{r}}_1}(\mathbf{1}) $ $ x_2(\overline{\mathbf{r}}_6)\leftarrow x_2+\overline{\mathbf{r}}_1 $ [0, 1, 1, 0] $ \mathfrak{R}_{4, \overline{\mathbf{r}}_8} $ $ x_4(\overline{\mathbf{r}}_{10})\leftarrow x_4+\overline{\mathbf{r}}_8 $ [3, 1, 2, 6]
    $ D_3=[\mathbf{1},\mathbf{ \pmb{\mathsf{ α}}^{-1}},\mathbf{1},\mathbf{1}] $ $ x_2(\overline{\mathbf{r}}_{6}^1)\leftarrow \mathbf{ \pmb{\mathsf{ α}}^{-1}} x_2 $ [0, 2, 1, 0] $ D_7=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1}] $ - [3, 1, 2, 6]
    $ \mathfrak{R}_{1, \overline{\mathbf{r}}_5}(\mathbf{ \pmb{\mathsf{ α}}}) $ $ x_1(\overline{\mathbf{r}}_7)\leftarrow x_1+\alpha\overline{\mathbf{r}}_5 $ [3, 1, 1, 0] $ \mathfrak{R}_{2, \overline{\mathbf{r}}_9^1}(\mathbf{1}) $ $ x_2(\overline{\mathbf{r}}_{11})\leftarrow x_2+ \overline{\mathbf{r}}_9^1 $ [3, 6, 2, 6]
    $ D_4=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1}] $ - [3, 1, 1, 0] $ D_8=[\mathbf{1},\mathbf{1},\mathbf{1},\mathbf{1}] $ - [3, 6, 2, 6]
    $ \mathfrak{R}_{3, \overline{\mathbf{r}}_6}(\mathbf{1}) $ $ x_3(\overline{\mathbf{r}}_8)\leftarrow x_3+\overline{\mathbf{r}}_6 $ [3, 1, 2, 0] $ \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}}(\mathbf{1}) $ $ x_1(\overline{\mathbf{r}}_{12})\leftarrow x_1+\overline{\mathbf{r}}_{11} $ [7, 6, 2, 6]
     | Show Table
    DownLoad: CSV

    Table 6.  The depth calculation of the path $ \mathfrak{P}_1 $

    Copy operation Transformation Depth Copy operation Transformation Depth
    $ \mathfrak{R}_{3, \overline{\mathbf{r}}_4} (\mathbf{1}) $ $ x_3(\overline{\mathbf{r}}_5)\leftarrow x_3+\overline{\mathbf{r}}_4 $ [0, 0, 1, 0] $ \mathfrak{R}_{4, \overline{\mathbf{r}}_7}(\mathbf{1}) $ $ x_4(\overline{\mathbf{r}}_9)\leftarrow x_4+\overline{\mathbf{r}}_7 $ [2, 1, 2, 3]
    $ \mathfrak{R}_{2, \overline{\mathbf{r}}_1}(\mathbf{1}) $ $ x_2(\overline{\mathbf{r}}_6)\leftarrow x_2+\overline{\mathbf{r}}_1 $ [0, 1, 1, 0] $ \mathfrak{R}_{4, \overline{\mathbf{r}}_8}(\mathbf{1}) $ $ x_4(\overline{\mathbf{r}}_{10})\leftarrow x_4+\overline{\mathbf{r}}_8 $ [2, 1, 2, 4]
    $ \mathfrak{R}_{1, \overline{\mathbf{r}}_5}(\mathbf{1}) $ $ x_1(\overline{\mathbf{r}}_7)\leftarrow x_1+\overline{\mathbf{r}}_5 $ [2, 1, 1, 0] $ \mathfrak{R}_{2, \overline{\mathbf{r}}_9}(\mathbf{1}) $ $ x_2(\overline{\mathbf{r}}_{11})\leftarrow x_2+\overline{\mathbf{r}}_9 $ [2, 4, 2, 4]
    $ \mathfrak{R}_{3, \overline{\mathbf{r}}_6}(\mathbf{1}) $ $ x_3(\overline{\mathbf{r}}_8)\leftarrow x_3+\overline{\mathbf{r}}_6 $ [2, 1, 2, 0] $ \mathfrak{R}_{1, \overline{\mathbf{r}}_{11}}(\mathbf{1}) $ $ x_1(\overline{\mathbf{r}}_{12})\leftarrow x_1+\overline{\mathbf{r}}_{11} $ [5, 4, 2, 4]
     | Show Table
    DownLoad: CSV

    Table 7.  $ 4\times 4 $ MDS matrices with low depth and cost

    Depth Cost 1st row 2nd row 3rd row 4th row
    5 [10,24] $ 35 $ $ [A+I, A, A^{-1}+I, A^{-1}] $ $ [A+I,I,A,A] $ $ [A, A, A^{-1}, A^{-1}+I] $ $ [I, A+I, A, A+I] $
    5 $ 35 $ $ [A^{-1}+I, A^{-1}, A+I, A] $ $ [A^{-1}+I, I, A^{-1} ,A^{-1}] $ $ [A^{-1}, A^{-1} ,A, A+I] $ $ [I, A^{-1}+I ,A^{-1}, A^{-1}+I] $
    4 [10] $ 37 $ $ [A+I,I,A+I,A] $ $ [A^2+A,A^2,I,I] $ $ [I,I,A,A+I] $ $ [A^2,A^2+A,I,A+I] $
    4 $ 37 $ $ [A+I,I,A^2+I,A^{-1}+A+I] $ $ [A^{-1}+A,A^{-1},I,I] $ $ [I,I,A^2 ,A^{-1}+A^2] $ $ [A^{-1}+A+I,A^{-1}+I,I,A^{-1}+I] $
    4 $ 37 $ $ [A^{-1}+I,I,A^{-1}+A^2,A^{-1}+A^2+I] $ $ [A^{-1}+I,A^{-1},I,I] $ $ [I,I,A^2,A^2+I] $ $ [A^{-1}+A+I,A^{-1}+A,I,A+I] $
    4 $ 37 $ $ [A^{-1}+I,I,A^{-1}+I,A^{-1}] $ $ [A^{-1}+A^2,A^2,I,I] $ $ [I,I,A^{-1},A^{-1}+I] $ $ [A^2,A^{-1}+A^2,I,A^{-1}+I] $
    3 [10] $ 41 $ $ [A+I, I ,A ,A] $ $ [I, A+I, A+I, I] $ $ [I ,I, A^{-1}+A^2, A^{-1}] $ $ [A^{-1}, A, I, A^{-1}+I] $
    3 $ 41 $ $ [A^{-1}+I ,I, A^{-1}, A^{-1}] $ $ [I, A^{-1}+I, A^{-1}+I, I] $ $ [I, I, A^{-1}+A, A] $ $ [A, A^{-1}, I, A+I] $
    5 [10,24] $ 67 $ $ [ B+I , B , B^7+B+I , B^7+B ] $ $ [ B+I , I , B , B ] $ $ [ B , B , B^7+B , B^7+B+I ] $ $ [ I , B+I , B , B+I ] $
    5 $ 67 $ $ [ B^7+B+I , B^7+B , B+I , B ] $ $ [ B^7+B+I , I , B^7+B , B^7+B ] $ $ [ B^7+B , B^7+B , B , B+I ] $ $ [ I , B^7+B+I , B^7+B , B^7+B+I ] $
    4 [10] $ 69 $ $ [ B+I , I , B+I , B ] $ $ [ B^2+B , B^2 , I , I ] $ $ [ I , I , B , B+I ] $ $ [ B^2 , B^2+B , I , B+I ] $
    4 $ 69 $ $ [ B+I , I , B^2+I , B^7+B^2+B+I ] $ $ [ B^7 , B^7+B , I , I ] $ $ [ I , I , B^2 , B^7+B^2+B ] $ $ [ B^7+I , B^7+B+I , I , B^7+B+I ] $
    4 $ 69 $ $ [ B^7+B+I , I , B^7+B^2+B , B^7+B^2+B+I ] $ $ [ B^7+B+I , B^7+B , I , I ] $ $ [ I , I , B^2 , B^2+I ] $ $ [ B^7+I , B^7 , I , B+I ] $
    4 $ 69 $ $ [ B^7+B+I , I , B^7+B+I , B^7+B ] $ $ [ B^7+B^2+B , B^2 , I , I ] $ $ [ I , I , B^7+B , B^7+B+I ] $ $ [ B^2 , B^7+B^2+B , I , B^7+B+I ] $
    3 [10] $ 77 $ $ [ B+I , I , B , B ] $ $ [ I , B+I , B+I , I ] $ $ [ I , I , B^7 , B^7+B ] $ $ [ B^7+B , B , I , B^7+B+I ] $
    3 $ 77 $ $ [ B^7+B+I , I , B^7+B , B^7+B ] $ $ [ I , B^7+B+I , B^7+B+I , I ] $ $ [ I , I , B^7 , B ] $ $ [ B , B^7+B , I , B+I ] $
    Where $A$ and $B$ are the companion matrices of the minimal polynomials $x^4+x+1$ and $x^8+x^2+1$, respectively.
     | Show Table
    DownLoad: CSV

    Table 8.  The statistical results

    Depth Cost Number of MDS matrices Depth Cost Number of MDS matrices
    - 35 52 - 67 52
    5 35 2 5 67 2
    4 37 4 4 69 4
    3 41 2 3 77 2
     | Show Table
    DownLoad: CSV
    • Related Papers
  • Cited by
  • References
  • [1] J. P. Aumasson, L. Henzen, W. Meier et al., QUARK: A lightweight hash, Cryptographic Hardware and Embedded Systems-CHES 2010, 6225 (2010), 1-15. doi: 10.1007/978-3-642-15031-9_1.
    [2] S. Banik, Y. Funabiki and T. Isobe, More results on shortest linear programs, Advances in Information and Computer Security-IWSEC 2019, 11689 (2019), 109-128. doi: 10.1007/978-3-030-26834-3_7.
    [3] A. Bogdanov, M. Knezevic, G. Leander et al., SPONGENT: A lightweight hash function, Cryptographic Hardware and Embedded Systems-CHES 2011, Lecture Notes in Computer Science, 6917 (2011), 312-325. doi: 10.1007/978-3-642-23951-9_21.
    [4] A. Bogdanov, L. R. Knudsen, G. Leander, C. Paar, A. Poschmann, M. J. B. Robshaw, Y. Seurin and C. Vikkelsoe, PRESENT: An ultra-lightweight block cipher, Cryptographic Hardware and Embedded Systems-CHES 2007, 4727 (2007), 450-466. doi: 10.1007/978-3-540-74735-2_31.
    [5] J. Boyar, P. Matthews and R. Peralta, On the shortest linear straight-line program for computing linear forms, Mathematical Foundations of Computer Science 2008, 33rd International Symposium, In MFCS 2008, 5162 (2008), 168-179. doi: 10.1007/978-3-540-85238-4_13.
    [6] J. Boyar, P. Matthews and R. Peralta, Logic minimization techniques with applications to cryptology,, J. Cryptology, 26 (2013), 280-312.  doi: 10.1007/s00145-012-9124-7.
    [7] J. Daemen and V. Rijmen, The wide trail design strategy, Cryptography and Coding, 8th IMA International Conference, 2260 (2001), 222-238. doi: 10.1007/3-540-45325-3_20.
    [8] J. Daemen and V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard, Information Security and Cryptography, Springer, 2002. doi: 10.1007/978-3-662-04722-4.
    [9] C. De Cannière, O. Dunkelman and M. Knezevic, KATAN and KTANTAN - A family of small and efficient hardware-oriented block ciphers, Cryptographic Hardware and Embedded Systems-CHES 2009, 5747 (2009), 272-288. doi: 10.1007/978-3-642-04138-9_20.
    [10] S. Duval and G. Leurent, MDS matrices with lightweight circuits, IACR Trans. Symmetric Cryptol., 2018 (2018), 48-78.  doi: 10.46586/tosc.v2018.i2.48-78.
    [11] J. Guo, T. Peyrin and A. Poschmann, The PHOTON family of lightweight hash functions, Advances in Cryptology-CRYPTO 2011, 6841 (2011), 222-239. doi: 10.1007/978-3-642-22792-9_13.
    [12] K. C. Gupta and I. G. Ray, On constructions of involutory MDS matrices, Progress in Cryptology-AFRICACRYPT 2013, 7918 (2013), 43-60. doi: 10.1007/978-3-642-38553-7_3.
    [13] H. M. Heys and S. E. Tavares, The design of substitution-permutation networks resistant to differential and linear cryptanalysis, CCS'94, Proceedings of the 2nd ACM Conference on Computer and Communications Security, (1994), 148-155.
    [14] J. Jean, T. Peyrin, S. M. Sim and J. Tourteaux, Optimizing implementations of lightweight building blocks, IACR Trans. Symmetric Cryptol., 2017 (2017), 130-168.  doi: 10.46586/tosc.v2017.i4.130-168.
    [15] K. Khoo, T. Peyrin, A. T. Poschmann et al., FOAM: Searching for hardware-optimal SPN structures and components with a fair comparison, Cryptographic Hardware and Embedded Systems-CHES 2014, 8731 (2014), 433-450. doi: 10.1007/978-3-662-44709-3_24.
    [16] T. Kranz, G. Leander, K. Stoffelen and F. Wiemer, Shorter linear straight-line programs for MDS matrices, IACR Trans. Symmetric Cryptol., 2017 (2017), 188-211.  doi: 10.46586/tosc.v2017.i4.188-211.
    [17] S. Li, S. Sun, C. Li, Z. Wei and L. Hu, Constructing low-latency involutory MDS matrices with lightweight circuits, IACR Trans. Symmetric Cryptol., 2019 (2019), 84-117.  doi: 10.46586/tosc.v2019.i1.84-117.
    [18] Y. Li and M. Wang, On the construction of lightweight circulant involutory MDS matrices, Fast Software Encryption 2016, 9783 (2016), 121-139.  doi: 10.1007/978-3-662-52993-5_7.
    [19] M. Liu and S. M. Sim, Lightweight MDS generalized circulant matrices, Fast Software Encryption 2016, 9783 (2016), 101-120.  doi: 10.1007/978-3-662-52993-5_6.
    [20] F. J. MacWilliams and N. J. A. Sloane, The theory of error correcting codes, North-Holland Mathematical Library, Amsterdam-New York Oxford: North-Holland Publishing Company, 16 (1977), 317-329. 
    [21] C. Paar, Optimized arithmetic for reed-solomon encoders, In: Proceedings of IEEE International Symposium on Information Theory, (1997), 250-250. doi: 10.1109/ISIT.1997.613165.
    [22] B. Ray, S. Douglas, S. Jason et al., The SIMON and SPECK families of lightweight block ciphers, Cryptology ePrint Archive, Report (2013), 414-414. Available from: http://eprint.iacr.org/2013/404.
    [23] S. M. Sim, K. Khoo, F. Oggier and T. Peyrin, Lightweight MDS involution matrices, In: Fast Software Encryption 2015, 9054 (2015), 471-493. doi: 10.1007/978-3-662-48116-5_23.
    [24] S. Wang, Y. Li, S. Tian et al., Four by four MDS matrices with the fewest XOR gates based on words, Advances in Mathematics of Communications, 2021. doi: 10.3934/amc.2021025.
    [25] Z. Xiang, X. Zeng, D. Lin, Z. Bao and S. Zhang, Optimizing implementations of linear layers, IACR Trans. Symmetric Cryptol., 2020 (2020), 120-145.  doi: 10.13154/tosc.v2020.i2.120-145.
    [26] Y. Yang, X. Zeng and S. Wang, Construction of lightweight involutory MDS matrices, Des. Codes Cryptogr., 89 (2021), 1453-1483.  doi: 10.1007/s10623-021-00879-3.
    • Access History
  • 加载中

Figures(2)

Tables(8)

PDF
XML
Export Citation
SHARE
Email to a Friend

Article Metrics

HTML views(2013) PDF downloads(53) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return
    Site map Copyright © 2025 American Institute of Mathematical Sciences