Design and Implementation of Generic Low-entropy High-order Composite Field Based Masking Scheme
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摘要:
通过对基于复合域S-box构造算法的深入研究，该文提出一种低面积复杂度的通用低熵高阶掩码算法。在有限域GF(24)上引入低熵掩码思想，并采用部分模块复用设计，有效降低了基于复合域S-box求逆运算的乘法数量。该算法能够适用于由求逆运算构成的任意分组加密算法，进一步将本方案应用于分组加密算法高级加密标准(AES)，给出了详细的综合仿真结果并进行了版图面积优化，较传统的掩码方案相比有效减少了逻辑资源的使用，此外，对其安全性进行了理论验证。
Abstract:Based on the in-depth research on the S-box constitution arithmetic of composite, an area optimized generic low-entropy higher-order masking scheme is proposed in this paper. The low entropy masking method is introduced on GF(24), and the partial module reusing design is adopted, which reduces effectively the number of multiplications based on the S-box inversion operation of the composite. The algorithm can be applied to any order masking scheme of arbitrary S-box composed of inversion operation. This scheme is applied to AES, gives detailed simulation results and optimizes the layout area, compared with the traditional masking scheme, reduces effectively the use of logical resources. In addition, the security is theoretically proved.
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Key words:
- High-order masking /
- Composite arithmetic /
- S-box /
- Low entropy /
- Advanced Encryption Standard(AES)
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表 1 低熵通用高阶掩码算法
算法1 低熵通用高阶掩码算法 输入：经掩码值$x = a + {m_1} + {m_2} + ··· + {m_d}$，掩码值${m_1},{m_2}, ··· ,{m_d}$ 输出：输入值的求逆${a^{ - 1}} + {m_1} + {m_2} + ··· + {m_d}$ (1) 通过同构矩阵$ \delta $，将有限域${\rm{GF}}({2^k})$上的输入值$x,$m1, m2, ··· ,md分别映射到有限域${\rm{GF}}({2^n})$上，$({x_h},{x_l}){\rm{ }} \leftarrow x$; $({m_{h1}},{m_{l1}}) \leftarrow {m_1}$;
$({m_{h2}},{m_{l2}}) \leftarrow {m_2}$; ··· ; $({m_{hd}},{m_{ld}}) \leftarrow {m_d}$；(2) 将有限域${\rm{GF}}({2^k})$上的求逆运算转化成有限域${\rm{GF}}({2^n})$上的加法、乘法，求逆运算； (3) 利用有限域${\rm{GF}}({2^n})$上的运算求取$d$的掩码防护值$d + {m_{h1} } + {m_{h2} } + ··· + {m_{hd} }$，$\begin{align} d + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}} = & {f_d}({x_h},({x_l} + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}} + {m_{l1}} + {m_{l2}} + ··· + {m_{ld}}),\\&({x_h} + {x_l} + {m_{l1}} + {m_{l2}} + ··· + {m_{ld}}),{m_{h1}},{m_{h2}}, ··· ,{m_{hd}},{m_{l1}},{m_{l2}}, ··· ,{m_{ld}},{{\rm{P}}_0})\\ =& {a_h}^2 \times {{\rm{P}}_0} + {a_h} \times {a_l} + {a_l}^2 + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}};\end{align}$ (4) 在有限域${\rm{GF}}({2^n})$上对$d + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}}$进行掩码求逆，求逆结果为${d^{ - 1}} + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}}$； (5) 利用有限域${\rm{GF}}({2^n})$上的运算求取${x_h},{x_l}$的掩码防护值${x_h}^\prime + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}}$, ${x_l}^\prime + {m_{l1} } + {m_{l2} } + ··· + {m_{ld} }$, $\begin{align}\quad\quad\ \; {x_h}^\prime + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}} =& {f_{bh}}({x_h},({d^{ - 1}} + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}}),\\&({x_h} + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}} + {d^{ - 1}} + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}}){\rm{ }},{m_{h1}},{m_{h2}}, ··· ,{m_{hd}})\\ = & {a_h} \times {d^{ - 1}} + {m_{h1}} + {m_{h2}} + ··· + {m_{hd}};\end{align}$$\begin{align}\quad\quad\ \; {x_l}^\prime + {m_{l1}} + {m_{l2}} + ··· + {m_{ld}} =& {f_{bl}}({x_h},({d^{ - 1}} + {m_{h1}} + {m_{h{\rm{2}}}} + ··· + {m_{hd}}), ({x_l} + {m_{h1}} + {m_{h{\rm{2}}}} + ··· + {m_{hd}} + {m_{l1}} + {m_{l2}} + ··· + {m_{ld}}),\\&({x_h} + {x_l} + {m_{l1}} + {m_{l2}} + ··· + {m_{ld}}) ,{m_{h1}},{m_{h2}}, ··· ,{m_{hd}},{m_{l1}},{m_{l2}}, ··· ,{m_{ld}})\\ =& ({a_h} + {a_l}) \times {d^{ - 1}} + {m_{l1}} + {m_{l2}} + ··· + {m_{ld}};\end{align}$ (6) 通过同构逆矩阵${{ \delta} ^{ - 1} }$，将有限域${\rm{GF}}({2^n})$上的求逆结果映射回有限域${\rm{GF}}({2^k})$上，得到有限域${\rm{GF}}({2^k})$上求逆结果
${a^{ - 1}} + {m_1} + {m_2} + {\rm{ }} ··· + {m_d}$。
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表 2 AES低熵通用高阶掩码算法
算法2 有限域${\rm{GF}}({2^k})$上对掩码单元$a' = a + {m_1} + {m_2}$求逆 输入：$(x = a + {m_1} + {m_2},{m_1},{m_2}) \in {\rm{GF}}({2^k})$ 输出：$(x' = {a^{ - 1}} + {m_1} + {m_2})$ (1) 映射${ \delta} ({m_1}) \to ({m_{h1} },{m_{l1} }) \in {\rm{GF} }({2^n})$，
映射$\delta ({m_2}) \to ({m_{h2}},{m_{l2}}) \in {\rm{GF}}({2^n})$；(2) 映射$\delta (x) \to ({x_h},{x_l}) \in {\rm{GF}}({2^n})$，即$\{ ({x_h},{x_l}) = ({a_h} + {m_{h1}} + {m_{h2}},{a_h} + {m_{l1}} + {m_{l2}})\} $；
$\begin{align}(3) \ d + {m_{h1}} + {m_{h2}} =& {({x_h})^2}{{\rm{P}}_0} + {m_h}{_1^2}{{\rm{P}}_0} + {m_h}{_2^2}{{\rm{P}}_0} + {({x_l} + {m_{h1}} + {m_{h2}} + {m_{l1}} + {m_{l2}})^2} + {x_h}({x_l} + {m_{h1}} + {m_{h2}} + {m_{l1}} + {m_{l2}})\\ &+ ({x_h} + {x_l} + {m_{l1}} + {m_{l2}})({m_{h1}} + {m_{h2}}) + {m_h}{_1^2} + {m_h}{_2^2} + {m_{h1}} + {m_{h2}};\end{align}$(4) ${d^{ - 1} } + {m_{h1} } + {m_{h2} } =\text{算法}2.(d + {m_{h1} } + {m_{h2} },{m_{h1} },{m_{h2} })$； (5) ${x_h}^\prime + {m_{h1}} + {m_{h2}} = {x_h}({d^{ - 1}} + {m_{h1}} + {m_{h2}}) + ({x_h} + {d^{ - 1}} + {m_{h1}} + {m_{h2}} + {m_{h1}} + {m_{h2}})({m_{h1}} + {m_{h2}}) + {m_{h1}} + {m_{h2}}$；
$\begin{align}(6) \ {x_l}^\prime + {m_{l1}} + {m_{l2}} =& {x_h}({d^{ - 1}} + {m_{h1}} + {m_{h2}}) + ({x_l} + {m_{h1}} + {m_{h2}} + {m_{l1}} + {m_{l2}})({d^{ - 1}} + {m_{h1}} + {m_{h2}}) + ({x_h} + {x_l} + {m_{l1}} + {m_{l2}})({m_{h1}} \\&+ {m_{h2}}) + {m_h}{_1^2} + {m_h}{_2^2} + {m_{l1}} + {m_{l2}}{text{；}}\end{align}$(7) 映射${\delta ^{ - 1}}({x_h}^\prime + {m_{h1}} + {m_{h2}},{x_l}^\prime + {m_{l1}} + {m_{l2}}) \to {\rm{ }}({a^{ - 1}} + {m_1} + {m_2})$。
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表 5 不同AES方案实现结果对比方案
思想 总的逻辑单元 组合逻辑 总的寄存器 非掩码 23890(21%) 19811(17%) 10769(9%) Oswald[12]
1阶掩码45549(40%)
(90.7%)40368(35%)
(103.8%)16036(14%)
(48.9%)汪鹏君[17]
1阶掩码42161(37%)
(76.5%)36584(32%)
(84.7%)13780(12%)
(28%)Ahn[18]
1阶掩码42087(37%)
(76.2%)36510(32%)
(84.3%)12820(11%)
(19%)本文方法
1阶掩码38456(34%)
(60.9%)32879(28%)
(66.1%)12820(11%)
(19%%)本文方法
2阶掩码44475(39%)
(86.1%)38282(33%)
(93.2%)18980(17%)
(76.2%%)
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表 6 本方案S-box不同掩码阶数的综合结果
掩码阶数 逻辑名称 逻辑面积(μm2) 合计(μm2) 非掩码S-box 组合逻辑，缓冲器/反相器逻辑，非组合逻辑 221, 14, 46 268 1阶掩码S-box 组合逻辑，缓冲器/反相器逻辑，非组合逻辑 489, 19, 88 577 2阶掩码S-box 组合逻辑，缓冲器/反相器逻辑，非组合逻辑 551, 22, 88 639
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表 7 本方案AES不同掩码阶数的综合结果
掩码阶数 逻辑名称 逻辑面积(μm2) 合计(μm2) 非掩码AES 组合逻辑，缓冲器/反相器逻辑，非组合逻辑 14484, 586, 53834 67518 1阶掩码AES 组合逻辑，缓冲器/反相器逻辑，非组合逻辑 53626, 2888, 59614 113241 2阶掩码AES 组合逻辑，缓冲器/反相器逻辑，非组合逻辑 116797, 4594, 100564 217361
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