SIMON類非線性函數(shù)的線性性質(zhì)研究
doi: 10.11999/JEIT200999
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戰(zhàn)略支援部隊信息工程大學(xué) 鄭州 450001
基金項目: 國家自然科學(xué)基金(61572516)
Research on Linear Properties of SIMON Class Nonlinear Function
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Strategic Support Forces Information Engineering University, Zhengzhou 450001, China
Funds: The National Natural Science Foundation of China (61572516)
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摘要: SIMON算法是由美國國家安全局(NSA)在2013 年推出的一簇輕量級分組密碼算法,具有實現(xiàn)代價低、安全性能好等優(yōu)點,其輪函數(shù)采用了
$F(x) = (x < < < a){{\& }}(x < < < b) \oplus (x < < < c)$ 類型的非線性函數(shù)。該文研究了移位參數(shù)(a,b,c)一般化時SIMON類算法輪函數(shù)的線性性質(zhì),解決了這類非線性函數(shù)的Walsh譜分布規(guī)律問題,證明了其相關(guān)優(yōu)勢只可能取到${{0}}$ 或${2^{ - k}}$ ,其中$k \in Z$ 且${{0}} \le k \le \left\lfloor {{2^{ - 1}}n} \right\rfloor $ ,并且對于特定條件下的每一個$k$ ,都存在相應(yīng)的掩碼對使得相關(guān)優(yōu)勢等于${2^{ - k}}$ ,給出了相關(guān)優(yōu)勢取到${2^{ - 1}}$ 時的充分必要條件及掩碼對的計數(shù),給出了特定條件下非平凡相關(guān)優(yōu)勢取到最小值時的充分必要條件與掩碼對的計數(shù)。-
關(guān)鍵詞:
- SIMON算法 /
- 線性性質(zhì) /
- 循環(huán)移位 /
- S盒
Abstract: SIMON algorithm is a group of lightweight block cipher algorithms introduced by the National Security Agency (NSA) in 2013. It has the advantages of low implementation cost and good security performance. Its round function adopts$F(x) = (x < < < a){{\& }}(x < < < b) \oplus (x < < < c)$ type nonlinear function. In this paper, the linear properties of the round function of SIMON algorithm when the shift parameters (a, b, c) are generalized are studied. The problem of Walsh spectrum distribution of this kind of nonlinear function is solved, it is proved that the correlation advantage can only be equal to 0 or${2^{ - k}}$ , where$k \in Z$ and${{0}} \le k \le \left\lfloor {{2^{ - 1}}n} \right\rfloor $ , and for each k under specific conditions, there are corresponding mask pairs so that the correlation advantage is equal to${2^{ - k}}$ . The necessary and sufficient conditions for the correlation advantage to be equal to 1/2 and the count of mask pairs are given. And the necessary and sufficient conditions for the nontrivial correlation advantage to be equal to the minimum value and the count of mask pairs under specific conditions are also given.-
Key words:
- SIMON algorithm /
- Linear property /
- Cyclic shift /
- S-box
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表 1
${F_{abc}}(x)$ 相關(guān)優(yōu)勢計數(shù)表$ \left| \rho \right|$ 0 1 1/2 1/4 1/8 1/16 1/32 $F_{182}^8$ 48255 1 64 1280 8256 7680 0 $F_{051}^8$ 48255 1 64 1280 8256 7680 0 $F_{182}^9$ 207863 1 72 1728 15360 37120 0 $F_{051}^9$ 207863 1 72 1728 15360 37120 0 下載: 導(dǎo)出CSV
表 2 轉(zhuǎn)變成不相交2次型算法(算法1)
輸入:2次型布爾函數(shù)$f\left( x \right) = f\left( {{x_1},{x_2}, \cdots ,{x_n}} \right)$ 輸出:可逆矩陣${\boldsymbol{M}}$,不相交二次型$\hat f\left( x \right)$使得$\hat f\left( x \right){\rm{ = }}f\left( {x{\boldsymbol{M}}} \right)$ (1) /*初始化*/ (2) ${\boldsymbol{M}} \leftarrow {\boldsymbol{I}}$ /*${\boldsymbol{I}}$是$n \times n$的可逆矩陣*/ (3) $\hat f\left( x \right) \leftarrow f\left( {{x_1},{x_2}, \cdots ,{x_n}} \right)$ (4) $v \leftarrow {\rm{PickIndex} }\left( {\hat f} \right)$ (5) /*不相交化*/ (6) 當(dāng)$\sigma \left( {\hat f,{x_v}} \right) \ge 2$時,執(zhí)行 (7) $m \leftarrow \sigma \left( {\hat f,{x_v}} \right)$ /*$\hat f$中包含${x_v}$的2次項個數(shù)*/ (8) 在$\hat f$中找出所有的2次項${x_v}{x_{{t_i}}}$滿足${t_1} < {t_2} < \cdots < {t_m}$ (9) $\hat f \leftarrow {\rm{Substitute}}\left( {\hat f,{{\boldsymbol{I}}_{{t_1} \leftarrow {t_1},{t_2}, \cdots ,{t_m}}}} \right)$ (10) ${\boldsymbol{M}} \leftarrow {{\boldsymbol{I}}_{{t_1} \leftarrow {t_1},{t_2}, \cdots ,{t_m}}} \cdot {\boldsymbol{M}}$ (11) 如果$\sigma \left( {\hat f,{x_{{t_1}}}} \right) \ge 2$,執(zhí)行 (12) $k \leftarrow \sigma \left( {\hat f,{x_{{t_1}}}} \right)$ (13) 在$\hat f$中找出所有的2次項${x_{{t_1}}}{x_{{s_i}}}$滿足
${s_1} < {s_2} < \cdots < {s_m}$,下載: 導(dǎo)出CSV
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