{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear codes for SSS-based polynomial masking\n", "\n", "### The SSS-based masking is a special case of GCM (Generalized Code-based masking)\n", "### $n=3$ shares, $t=1$, $\\ell=4$ bits: (3,1)-SSS (Shamir's Secret Sharing)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Parameters:\n", "\n", "- $Z=(X + \\alpha_1Y_1, X + \\alpha_2Y_1, X + \\alpha_3Y_1)=X\\mathbf{G} + Y\\mathbf{H}$ where $X, Y=(Y_1)$ and $Z$ are the sensitive variable, a mask and the protected variable, respectively, where $\\alpha_i$ for $1\\leq i\\leq 3$ are three public points in SSS-scheme\n", "- $\\mathbf{G} = [1, 1, 1]$ and $\\mathbf{H} = [\\alpha_1, \\alpha_2, \\alpha_3]$ are two generator matrices of codes $\\mathcal{C}$ and $\\mathcal{D}$, resp.\n", "- $\\alpha_i\\in \\mathbb{F}_{2^\\ell}\\backslash\\{0\\}$ and $\\alpha_1\\neq\\alpha_2\\neq\\alpha_3$, thus there are ${15}\\choose{3}$=455 linear codes for (3,1)-SSS\n", "- Each nonzero element over $\\mathbb{F}_{2^\\ell}$ can be denoted as $\\alpha^i$ where $i\\in\\{0, 1, \\ldots, 14\\}$, the corresponding irreducible polynomial is $g(\\alpha) = \\alpha^4 + \\alpha +1$\n", "- Due to equivalence of linear codes, we simplify the enumeration by choosing $(\\alpha_1, \\alpha_2, \\alpha_3)=(\\alpha^i, \\alpha^j,\\alpha^k)$ where $i=0$ and $0\n", " #T_8d379dee_117e_11eb_bc3d_8c16459d77ae th {\n", " font-size: 14px;\n", " text-align: left;\n", " font-weight: bold;\n", " background-color: #E0E0E0;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77ae td {\n", " font-size: 13px;\n", " text-align: left;\n", " min-width: 80px;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77ae caption {\n", " font-size: 16px;\n", " text-align: center;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow0_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow0_col3 {\n", " background-color: #ff8181;\n", " color: #000000;\n", " 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background-color: #ffe5ff;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow43_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow43_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow44_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow44_col3 {\n", " background-color: #ffd7d7;\n", " color: #000000;\n", " background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow45_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow45_col3 {\n", " background-color: #ffc9c9;\n", " color: #000000;\n", " background-color: #901d90;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow46_col2 {\n", " background-color: #ffe5e5;\n", 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background-color: #ffe5ff;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow50_col2 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow50_col3 {\n", " background-color: #ff0e0e;\n", " color: #f1f1f1;\n", " background-color: #f8d8f8;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow51_col2 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow51_col3 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " background-color: #ffe5ff;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow52_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow52_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow53_col2 {\n", " background-color: #ffe5e5;\n", 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background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow57_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow57_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow58_col2 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow58_col3 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " background-color: #ffe5ff;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow59_col2 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow59_col3 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " background-color: #ffe5ff;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow60_col2 {\n", " background-color: #ffe5e5;\n", 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background-color: #901d90;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow64_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow64_col3 {\n", " background-color: #ffd7d7;\n", " color: #000000;\n", " background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow65_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow65_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow66_col2 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow66_col3 {\n", " background-color: #ff0000;\n", " color: #f1f1f1;\n", " background-color: #ffe5ff;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow67_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow67_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow68_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow68_col3 {\n", " background-color: #ffd7d7;\n", " color: #000000;\n", " background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow69_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow69_col3 {\n", " background-color: #ffc9c9;\n", " color: #000000;\n", " background-color: #901d90;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow70_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow70_col3 {\n", " background-color: #ffc9c9;\n", " color: #000000;\n", " background-color: #901d90;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow71_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow71_col3 {\n", " background-color: #ffd7d7;\n", " color: #000000;\n", " background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow72_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow72_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow73_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow73_col3 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " background-color: #800080;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow74_col2 {\n", " background-color: #ffe5e5;\n", 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background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow78_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow78_col3 {\n", " background-color: #ffd7d7;\n", " color: #000000;\n", " background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow79_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow79_col3 {\n", " background-color: #ffd7d7;\n", " color: #000000;\n", " background-color: #880e88;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow80_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow80_col3 {\n", " background-color: #ffc9c9;\n", " color: #000000;\n", " background-color: #901d90;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow81_col2 {\n", " background-color: #ffe5e5;\n", 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background-color: #901d90;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow85_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow85_col3 {\n", " background-color: #ffbaba;\n", " color: #000000;\n", " background-color: #982b98;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow86_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow86_col3 {\n", " background-color: #ffbaba;\n", " color: #000000;\n", " background-color: #982b98;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow87_col2 {\n", " background-color: #ffe5e5;\n", " color: #000000;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow87_col3 {\n", " background-color: #ffbaba;\n", " color: #000000;\n", " background-color: #982b98;\n", " color: #f1f1f1;\n", " } #T_8d379dee_117e_11eb_bc3d_8c16459d77aerow88_col2 {\n", " background-color: #ffe5e5;\n", 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" \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tab. I All linear codes for (3,1)-SSS-based masking with $n=3$ shares over $\\mathbb{F}_{2^4}$.
$\\alpha_2$ $\\alpha_3$ $d_{\\mathcal{D}}^\\perp$ $B_{d_{\\mathcal{D}}^\\perp}$ Weight Enumerators
0$\\alpha^{1}$$\\alpha^{2}$28[0, 1, 2, 8, 3, 10, 4, 28, 5, 50, 6, 50, 7, 62, 8, 35, 9, 6, 10, 6, 1, 3]
1$\\alpha^{1}$$\\alpha^{3}$26[0, 1, 2, 6, 3, 14, 4, 22, 5, 52, 6, 64, 7, 48, 8, 33, 9, 12, 10, 2, 11, 2, 1, 4]
2$\\alpha^{1}$$\\alpha^{4}$24[0, 1, 2, 4, 3, 16, 4, 23, 5, 48, 6, 72, 7, 48, 8, 23, 9, 16, 10, 4, 12, 1, 1, 5]
3$\\alpha^{1}$$\\alpha^{5}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 1, 6]
4$\\alpha^{1}$$\\alpha^{6}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 1, 7]
5$\\alpha^{1}$$\\alpha^{7}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 1, 8]
6$\\alpha^{1}$$\\alpha^{8}$23[0, 1, 2, 3, 3, 14, 4, 30, 5, 52, 6, 58, 7, 48, 8, 33, 9, 12, 10, 3, 11, 2, 1, 9]
7$\\alpha^{1}$$\\alpha^{9}$23[0, 1, 2, 3, 3, 13, 4, 30, 5, 56, 6, 58, 7, 42, 8, 33, 9, 16, 10, 3, 11, 1, 1, 10]
8$\\alpha^{1}$$\\alpha^{10}$23[0, 1, 2, 3, 3, 10, 4, 36, 5, 58, 6, 48, 7, 46, 8, 35, 9, 14, 10, 5, 1, 11]
9$\\alpha^{1}$$\\alpha^{11}$23[0, 1, 2, 3, 3, 10, 4, 36, 5, 58, 6, 48, 7, 46, 8, 35, 9, 14, 10, 5, 1, 12]
10$\\alpha^{1}$$\\alpha^{12}$24[0, 1, 2, 4, 3, 12, 4, 28, 5, 58, 6, 58, 7, 42, 8, 35, 9, 14, 10, 2, 11, 2, 1, 13]
11$\\alpha^{1}$$\\alpha^{13}$26[0, 1, 2, 6, 3, 12, 4, 26, 5, 52, 6, 60, 7, 52, 8, 29, 9, 12, 10, 6, 1, 14]
12$\\alpha^{1}$$\\alpha^{14}$28[0, 1, 2, 8, 3, 10, 4, 28, 5, 50, 6, 50, 7, 62, 8, 35, 9, 6, 10, 6, 2, 3]
13$\\alpha^{2}$$\\alpha^{3}$26[0, 1, 2, 6, 3, 12, 4, 26, 5, 52, 6, 60, 7, 52, 8, 29, 9, 12, 10, 6, 2, 4]
14$\\alpha^{2}$$\\alpha^{4}$24[0, 1, 2, 4, 3, 14, 4, 26, 5, 52, 6, 64, 7, 48, 8, 29, 9, 12, 10, 4, 11, 2, 2, 5]
15$\\alpha^{2}$$\\alpha^{5}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 2, 6]
16$\\alpha^{2}$$\\alpha^{6}$22[0, 1, 2, 2, 3, 16, 4, 30, 5, 48, 6, 64, 7, 48, 8, 25, 9, 16, 10, 6, 2, 7]
17$\\alpha^{2}$$\\alpha^{7}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 2, 8]
18$\\alpha^{2}$$\\alpha^{8}$22[0, 1, 2, 2, 3, 16, 4, 31, 5, 48, 6, 60, 7, 48, 8, 31, 9, 16, 10, 2, 12, 1, 2, 9]
19$\\alpha^{2}$$\\alpha^{9}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 2, 10]
20$\\alpha^{2}$$\\alpha^{10}$22[0, 1, 2, 2, 3, 13, 4, 34, 5, 56, 6, 52, 7, 42, 8, 37, 9, 16, 10, 2, 11, 1, 2, 11]
21$\\alpha^{2}$$\\alpha^{11}$22[0, 1, 2, 2, 3, 13, 4, 34, 5, 56, 6, 52, 7, 42, 8, 37, 9, 16, 10, 2, 11, 1, 2, 12]
22$\\alpha^{2}$$\\alpha^{12}$23[0, 1, 2, 3, 3, 12, 4, 34, 5, 52, 6, 54, 7, 52, 8, 29, 9, 12, 10, 7, 2, 13]
23$\\alpha^{2}$$\\alpha^{13}$24[0, 1, 2, 4, 3, 14, 4, 26, 5, 52, 6, 64, 7, 48, 8, 29, 9, 12, 10, 4, 11, 2, 2, 14]
24$\\alpha^{2}$$\\alpha^{14}$26[0, 1, 2, 6, 3, 14, 4, 22, 5, 52, 6, 64, 7, 48, 8, 33, 9, 12, 10, 2, 11, 2, 3, 4]
25$\\alpha^{3}$$\\alpha^{4}$24[0, 1, 2, 4, 3, 12, 4, 28, 5, 58, 6, 58, 7, 42, 8, 35, 9, 14, 10, 2, 11, 2, 3, 5]
26$\\alpha^{3}$$\\alpha^{5}$23[0, 1, 2, 3, 3, 12, 4, 34, 5, 52, 6, 54, 7, 52, 8, 29, 9, 12, 10, 7, 3, 6]
27$\\alpha^{3}$$\\alpha^{6}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 3, 7]
28$\\alpha^{3}$$\\alpha^{7}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 3, 8]
29$\\alpha^{3}$$\\alpha^{8}$21[0, 1, 2, 1, 3, 16, 4, 36, 5, 46, 6, 52, 7, 54, 8, 35, 9, 10, 10, 3, 11, 2, 3, 9]
30$\\alpha^{3}$$\\alpha^{9}$21[0, 1, 2, 1, 3, 17, 4, 34, 5, 44, 6, 58, 7, 54, 8, 29, 9, 12, 10, 5, 11, 1, 3, 10]
31$\\alpha^{3}$$\\alpha^{10}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 3, 11]
32$\\alpha^{3}$$\\alpha^{11}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 3, 12]
33$\\alpha^{3}$$\\alpha^{12}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 3, 13]
34$\\alpha^{3}$$\\alpha^{13}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 3, 14]
35$\\alpha^{3}$$\\alpha^{14}$24[0, 1, 2, 4, 3, 16, 4, 23, 5, 48, 6, 72, 7, 48, 8, 23, 9, 16, 10, 4, 12, 1, 4, 5]
36$\\alpha^{4}$$\\alpha^{5}$23[0, 1, 2, 3, 3, 10, 4, 36, 5, 58, 6, 48, 7, 46, 8, 35, 9, 14, 10, 5, 4, 6]
37$\\alpha^{4}$$\\alpha^{6}$22[0, 1, 2, 2, 3, 13, 4, 34, 5, 56, 6, 52, 7, 42, 8, 37, 9, 16, 10, 2, 11, 1, 4, 7]
38$\\alpha^{4}$$\\alpha^{7}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 4, 8]
39$\\alpha^{4}$$\\alpha^{8}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 9]
40$\\alpha^{4}$$\\alpha^{9}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 10]
41$\\alpha^{4}$$\\alpha^{10}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 11]
42$\\alpha^{4}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 12]
43$\\alpha^{4}$$\\alpha^{12}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 4, 13]
44$\\alpha^{4}$$\\alpha^{13}$22[0, 1, 2, 2, 3, 16, 4, 30, 5, 48, 6, 64, 7, 48, 8, 25, 9, 16, 10, 6, 4, 14]
45$\\alpha^{4}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 5, 6]
46$\\alpha^{5}$$\\alpha^{6}$23[0, 1, 2, 3, 3, 10, 4, 36, 5, 58, 6, 48, 7, 46, 8, 35, 9, 14, 10, 5, 5, 7]
47$\\alpha^{5}$$\\alpha^{7}$22[0, 1, 2, 2, 3, 13, 4, 34, 5, 56, 6, 52, 7, 42, 8, 37, 9, 16, 10, 2, 11, 1, 5, 8]
48$\\alpha^{5}$$\\alpha^{8}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 5, 9]
49$\\alpha^{5}$$\\alpha^{9}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 5, 10]
50$\\alpha^{5}$$\\alpha^{10}$316[0, 1, 3, 16, 4, 39, 5, 48, 6, 48, 7, 48, 8, 39, 9, 16, 12, 1, 5, 11]
51$\\alpha^{5}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 5, 12]
52$\\alpha^{5}$$\\alpha^{12}$21[0, 1, 2, 1, 3, 16, 4, 36, 5, 46, 6, 52, 7, 54, 8, 35, 9, 10, 10, 3, 11, 2, 5, 13]
53$\\alpha^{5}$$\\alpha^{13}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 5, 14]
54$\\alpha^{5}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 6, 7]
55$\\alpha^{6}$$\\alpha^{7}$23[0, 1, 2, 3, 3, 13, 4, 30, 5, 56, 6, 58, 7, 42, 8, 33, 9, 16, 10, 3, 11, 1, 6, 8]
56$\\alpha^{6}$$\\alpha^{8}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 6, 9]
57$\\alpha^{6}$$\\alpha^{9}$21[0, 1, 2, 1, 3, 17, 4, 34, 5, 44, 6, 58, 7, 54, 8, 29, 9, 12, 10, 5, 11, 1, 6, 10]
58$\\alpha^{6}$$\\alpha^{10}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 6, 11]
59$\\alpha^{6}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 6, 12]
60$\\alpha^{6}$$\\alpha^{12}$21[0, 1, 2, 1, 3, 17, 4, 34, 5, 44, 6, 58, 7, 54, 8, 29, 9, 12, 10, 5, 11, 1, 6, 13]
61$\\alpha^{6}$$\\alpha^{13}$22[0, 1, 2, 2, 3, 16, 4, 31, 5, 48, 6, 60, 7, 48, 8, 31, 9, 16, 10, 2, 12, 1, 6, 14]
62$\\alpha^{6}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 7, 8]
63$\\alpha^{7}$$\\alpha^{8}$23[0, 1, 2, 3, 3, 14, 4, 30, 5, 52, 6, 58, 7, 48, 8, 33, 9, 12, 10, 3, 11, 2, 7, 9]
64$\\alpha^{7}$$\\alpha^{9}$22[0, 1, 2, 2, 3, 16, 4, 31, 5, 48, 6, 60, 7, 48, 8, 31, 9, 16, 10, 2, 12, 1, 7, 10]
65$\\alpha^{7}$$\\alpha^{10}$21[0, 1, 2, 1, 3, 16, 4, 36, 5, 46, 6, 52, 7, 54, 8, 35, 9, 10, 10, 3, 11, 2, 7, 11]
66$\\alpha^{7}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 7, 12]
67$\\alpha^{7}$$\\alpha^{12}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 7, 13]
68$\\alpha^{7}$$\\alpha^{13}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 7, 14]
69$\\alpha^{7}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 14, 4, 30, 5, 52, 6, 58, 7, 48, 8, 33, 9, 12, 10, 3, 11, 2, 8, 9]
70$\\alpha^{8}$$\\alpha^{9}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 8, 10]
71$\\alpha^{8}$$\\alpha^{10}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 8, 11]
72$\\alpha^{8}$$\\alpha^{11}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 8, 12]
73$\\alpha^{8}$$\\alpha^{12}$21[0, 1, 2, 1, 3, 16, 4, 34, 5, 48, 6, 58, 7, 48, 8, 29, 9, 16, 10, 5, 8, 13]
74$\\alpha^{8}$$\\alpha^{13}$22[0, 1, 2, 2, 3, 13, 4, 34, 5, 56, 6, 52, 7, 42, 8, 37, 9, 16, 10, 2, 11, 1, 8, 14]
75$\\alpha^{8}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 13, 4, 30, 5, 56, 6, 58, 7, 42, 8, 33, 9, 16, 10, 3, 11, 1, 9, 10]
76$\\alpha^{9}$$\\alpha^{10}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 9, 11]
77$\\alpha^{9}$$\\alpha^{11}$22[0, 1, 2, 2, 3, 16, 4, 30, 5, 48, 6, 64, 7, 48, 8, 25, 9, 16, 10, 6, 9, 12]
78$\\alpha^{9}$$\\alpha^{12}$22[0, 1, 2, 2, 3, 15, 4, 32, 5, 50, 6, 58, 7, 48, 8, 31, 9, 14, 10, 4, 11, 1, 9, 13]
79$\\alpha^{9}$$\\alpha^{13}$22[0, 1, 2, 2, 3, 13, 4, 34, 5, 56, 6, 52, 7, 42, 8, 37, 9, 16, 10, 2, 11, 1, 9, 14]
80$\\alpha^{9}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 10, 4, 36, 5, 58, 6, 48, 7, 46, 8, 35, 9, 14, 10, 5, 10, 11]
81$\\alpha^{10}$$\\alpha^{11}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 10, 12]
82$\\alpha^{10}$$\\alpha^{12}$23[0, 1, 2, 3, 3, 15, 4, 28, 5, 50, 6, 64, 7, 48, 8, 27, 9, 14, 10, 5, 11, 1, 10, 13]
83$\\alpha^{10}$$\\alpha^{13}$23[0, 1, 2, 3, 3, 12, 4, 34, 5, 52, 6, 54, 7, 52, 8, 29, 9, 12, 10, 7, 10, 14]
84$\\alpha^{10}$$\\alpha^{14}$23[0, 1, 2, 3, 3, 10, 4, 36, 5, 58, 6, 48, 7, 46, 8, 35, 9, 14, 10, 5, 11, 12]
85$\\alpha^{11}$$\\alpha^{12}$24[0, 1, 2, 4, 3, 16, 4, 23, 5, 48, 6, 72, 7, 48, 8, 23, 9, 16, 10, 4, 12, 1, 11, 13]
86$\\alpha^{11}$$\\alpha^{13}$24[0, 1, 2, 4, 3, 14, 4, 26, 5, 52, 6, 64, 7, 48, 8, 29, 9, 12, 10, 4, 11, 2, 11, 14]
87$\\alpha^{11}$$\\alpha^{14}$24[0, 1, 2, 4, 3, 12, 4, 28, 5, 58, 6, 58, 7, 42, 8, 35, 9, 14, 10, 2, 11, 2, 12, 13]
88$\\alpha^{12}$$\\alpha^{13}$26[0, 1, 2, 6, 3, 14, 4, 22, 5, 52, 6, 64, 7, 48, 8, 33, 9, 12, 10, 2, 11, 2, 12, 14]
89$\\alpha^{12}$$\\alpha^{14}$26[0, 1, 2, 6, 3, 12, 4, 26, 5, 52, 6, 60, 7, 52, 8, 29, 9, 12, 10, 6, 13, 14]
90$\\alpha^{13}$$\\alpha^{14}$28[0, 1, 2, 8, 3, 10, 4, 28, 5, 50, 6, 50, 7, 62, 8, 35, 9, 6, 10, 6]
" ], "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.DataFrame({'$\\\\alpha_2$': alpha_all[alpha_2[:]], '$\\\\alpha_3$': alpha_all[alpha_3[:]], '$d_{\\mathcal{D}}^\\perp$': d_all[:], \n", " '$B_{d_{\\mathcal{D}}^\\perp}$': B_all[:], 'Weight Enumerators': d_dual_b[:]})\n", "\n", "pd.set_option('display.max_colwidth', 1000)\n", "pd.set_option('display.width', 800)\n", "(df.style\n", " .background_gradient(cmap=cm_1, subset=['$d_{\\mathcal{D}}^\\perp$','$B_{d_{\\mathcal{D}}^\\perp}$' ])\n", " .background_gradient(cmap=cm_2, subset=['$B_{d_{\\mathcal{D}}^\\perp}$' ])\n", " .set_caption('Tab. I All linear codes for (3,1)-SSS-based masking with $n=3$ shares over $\\mathbb{F}_{2^4}$.')\n", " .set_table_styles(styles))\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2. Optimal linear codes for (3,1)-SSS-based masking" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.1 Linear codes with $d_{\\mathcal{D}}^\\perp=3$\n", "\n", "We focus on the the linear codes with greater $d_{\\mathcal{D}}^\\perp$, which are better in the sense of side-channel resistance (from our paper)." ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:14:48.353129Z", "start_time": "2020-10-18T20:14:48.339321Z" } }, "outputs": [], "source": [ "# Finding the indices of d_C=6\n", "d_index = []\n", "d_index_alpha_2 = []\n", "d_index_alpha_3 = []\n", "d_D_perp = 3\n", "for i in range(len(d_dual_b)):\n", " if d_dual_b[i][2] == d_D_perp:\n", " d_index.append(i)\n", " d_index_alpha_2.append(pow_ind[i][0])\n", " d_index_alpha_3.append(pow_ind[i][1])\n", "\n", "d_index = np.array(d_index)\n", "d_index_alpha_2 = np.array(d_index_alpha_2)\n", "d_index_alpha_3 = np.array(d_index_alpha_3)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:14:50.060052Z", "start_time": "2020-10-18T20:14:50.052829Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "10\n", "[39 40 41 42 49 50 51 58 59 66]\n" ] } ], "source": [ "print(len(d_index))\n", "print(d_index)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:14:52.634721Z", "start_time": "2020-10-18T20:14:52.620205Z" } }, "outputs": [], "source": [ "def highlight(s, threshold, column):\n", " is_min = pd.Series(data=False, index=s.index)\n", " is_min[column] = (s.loc[column] <= threshold)\n", " return ['background-color: gold' if is_min.any() else '' for v in is_min]" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "ExecuteTime": { "end_time": "2020-10-18T20:14:57.863683Z", "start_time": "2020-10-18T20:14:57.714339Z" }, "scrolled": false }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tab. II Linear codes for (3,1)-SSS-based masking with $d_{\\mathcal{D}}^\\perp=3$.
$\\alpha_2$ $\\alpha_3$ $d_{\\mathcal{D}}^\\perp$ $B_{d_{\\mathcal{D}}^\\perp}$ Weight Enumerators
5$\\alpha^{5}$$\\alpha^{10}$316[0, 1, 3, 16, 4, 39, 5, 48, 6, 48, 7, 48, 8, 39, 9, 16, 12, 1, 5, 11]
0$\\alpha^{4}$$\\alpha^{8}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 9]
1$\\alpha^{4}$$\\alpha^{9}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 10]
2$\\alpha^{4}$$\\alpha^{10}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 11]
3$\\alpha^{4}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 4, 12]
4$\\alpha^{5}$$\\alpha^{9}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 5, 10]
6$\\alpha^{5}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 5, 12]
7$\\alpha^{6}$$\\alpha^{10}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 6, 11]
8$\\alpha^{6}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 6, 12]
9$\\alpha^{7}$$\\alpha^{11}$317[0, 1, 3, 17, 4, 38, 5, 44, 6, 52, 7, 54, 8, 33, 9, 12, 10, 4, 11, 1, 7, 12]
" ], "text/plain": [ "" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_4 = pd.DataFrame({'$\\\\alpha_2$': np.array(alpha_all)[d_index_alpha_2], '$\\\\alpha_3$': np.array(alpha_all)[d_index_alpha_3], '$d_{\\mathcal{D}}^\\perp$': \n", " d_all[d_index], '$B_{d_{\\mathcal{D}}^\\perp}$': B_all[d_index], 'Weight Enumerators': np.array(d_dual_b)[d_index]})\n", "df_4 = df_4.sort_values(by=['$B_{d_{\\mathcal{D}}^\\perp}$'], ascending=True)\n", "\n", "(df_4.style\n", " .apply(highlight, threshold=16, column=['$B_{d_{\\mathcal{D}}^\\perp}$'], axis=1)\n", " .background_gradient(cmap=cm_2, subset=['$B_{d_{\\mathcal{D}}^\\perp}$' ])\n", " .set_caption('Tab. II Linear codes for (3,1)-SSS-based masking with $d_{\\mathcal{D}}^\\perp=3$.')\n", " .set_table_styles(styles))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 2.2 Optimal codes for (3,1)-SSS-based masking\n", "\n", "As shown in our paper, the codes satifying two conditions are optimal:\n", "\n", "- Maximizing $d_{\\mathcal{D}}^\\perp$, here $\\max\\{d_{\\mathcal{D}}^\\perp\\} = 3$\n", "- Minimizing $B_{d_{\\mathcal{D}}^\\perp}$, here $\\min\\{B_{d_{\\mathcal{D}}^\\perp}\\} = 16$\n", "\n", "Note that we use two complementary metrics **SNR** (signal-to-noise ratio) and **MI** (mutual information) to assess the side-channel resistance of SSS-based masking with different codes." ] }, { "cell_type": "markdown", "metadata": { "scrolled": false }, "source": [ "As a result of Tab. II, we conclude that the optimal codes for (3,1)-SSS based masking are generated by $\\mathbf{H}=[\\alpha_1, \\alpha_2,\\alpha_3]$ where $(\\alpha_1, \\alpha_2,\\alpha_3)\\in\\{(\\alpha^0, \\alpha^5, \\alpha^{10})\\}$. Note that permutation on three public points does not change the codes due to equivalence.\n", "\n", "The generator matrix of the code is: \n", "$$\n", "\\mathbf{H}_{optimal}=\\left( \\begin{matrix} \\alpha^0 & \\alpha^5 & \\alpha^{10} \\end{matrix} \\right) \\in \\mathbb{F}_{2^4}^{1\\times 3} \n", "= \\left(\n", " \\begin{matrix}\n", " 1&0&0&0&0&1&1&0&1&1&1&0 \\\\\n", " 0&1&0&0&0&0&1&1&0&1&1&1 \\\\\n", " 0&0&1&0&1&1&0&1&1&1&1&1 \\\\\n", " 0&0&0&1&1&0&1&0&1&0&1&1 \n", " \\end{matrix} \n", "\\right) \\normalsize\\in \\mathbb{F}_2^{4\\times 12}\n", "$$ " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": false, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }