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SIMON類非線性函數(shù)的線性性質(zhì)研究

關(guān)杰 盧健偉

關(guān)杰, 盧健偉. SIMON類非線性函數(shù)的線性性質(zhì)研究[J]. 電子與信息學(xué)報, 2021, 43(11): 3359-3366. doi: 10.11999/JEIT200999
引用本文: 關(guān)杰, 盧健偉. SIMON類非線性函數(shù)的線性性質(zhì)研究[J]. 電子與信息學(xué)報, 2021, 43(11): 3359-3366. doi: 10.11999/JEIT200999
Jie GUAN, Jianwei LU. Research on Linear Properties of SIMON Class Nonlinear Function[J]. Journal of Electronics & Information Technology, 2021, 43(11): 3359-3366. doi: 10.11999/JEIT200999
Citation: Jie GUAN, Jianwei LU. Research on Linear Properties of SIMON Class Nonlinear Function[J]. Journal of Electronics & Information Technology, 2021, 43(11): 3359-3366. doi: 10.11999/JEIT200999

SIMON類非線性函數(shù)的線性性質(zhì)研究

doi: 10.11999/JEIT200999
基金項目: 國家自然科學(xué)基金(61572516)
詳細(xì)信息
    作者簡介:

    關(guān)杰:女,1974年生,教授,博士生導(dǎo)師,研究方向為密碼理論和密碼算法分析

    盧健偉:男,1997年生,碩士生,研究方向為對稱密碼設(shè)計與分析

    通訊作者:

    盧健偉 lujianwei1997@163.com

  • 中圖分類號: TN918.1

Research on Linear Properties of SIMON Class Nonlinear Function

Funds: The National Natural Science Foundation of China (61572516)
  • 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}}$${2^{ - k}}$,其中$k \in Z$${{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)。
  • 表  1  ${F_{abc}}(x)$相關(guān)優(yōu)勢計數(shù)表

    $ \left| \rho \right|$
    011/21/41/81/161/32
    $F_{182}^8$482551641280825676800
    $F_{051}^8$482551641280825676800
    $F_{182}^9$207863172172815360371200
    $F_{051}^9$207863172172815360371200
    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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    DONG Xiangzhong, GUAN Jie. Analysis on difffferential properties of the round function of SIMON family of block ciphers[J]. Journal of Cryptologic Research, 2015, 2(3): 207–216. doi: 10.13868/j.cnki.jcr.000072
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    DONG Xiangzhong, GUAN Jie. Linear properties of the round function of SIMON family of block ciphers[J]. 山東大學(xué)學(xué)報, 2015, 50(9): 49–54.
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出版歷程
  • 收稿日期:  2020-11-25
  • 修回日期:  2021-03-30
  • 網(wǎng)絡(luò)出版日期:  2021-05-06
  • 刊出日期:  2021-11-23

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