SIMON類(lèi)非線性函數(shù)的線性性質(zhì)研究

關(guān)杰; 盧健偉

doi:10.11999/JEIT200999

SIMON類(lèi)非線性函數(shù)的線性性質(zhì)研究

doi: 10.11999/JEIT200999 cstr: 32379.14.JEIT200999

關(guān)杰,
盧健偉^,

戰(zhàn)略支援部隊(duì)信息工程大學(xué) 鄭州 450001

基金項(xiàng)目: 國(guó)家自然科學(xué)基金(61572516)

詳細(xì)信息

作者簡(jiǎn)介:
關(guān)杰：女，1974年生，教授，博士生導(dǎo)師，研究方向?yàn)槊艽a理論和密碼算法分析

盧健偉：男，1997年生，碩士生，研究方向?yàn)閷?duì)稱(chēng)密碼設(shè)計(jì)與分析

通訊作者:
盧健偉　lujianwei1997@163.com

中圖分類(lèi)號(hào): TN918.1
計(jì)量
- 文章訪問(wèn)數(shù): 927
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- 被引次數(shù): 0
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

Research on Linear Properties of SIMON Class Nonlinear Function

Jie GUAN,
Jianwei LU^,

Strategic Support Forces Information Engineering University, Zhengzhou 450001, China

Funds: The National Natural Science Foundation of China (61572516)

摘要

摘要: SIMON算法是由美國(guó)國(guó)家安全局(NSA)在2013 年推出的一簇輕量級(jí)分組密碼算法，具有實(shí)現(xiàn)代價(jià)低、安全性能好等優(yōu)點(diǎn)，其輪函數(shù)采用了$F(x) = (x < < < a){{\& }}(x < < < b) \oplus (x < < < c)$類(lèi)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(shí)SIMON類(lèi)算法輪函數(shù)的線性性質(zhì)，解決了這類(lèi)非線性函數(shù)的Walsh譜分布規(guī)律問(wèn)題，證明了其相關(guān)優(yōu)勢(shì)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 $，并且對(duì)于特定條件下的每一個(gè)$k$，都存在相應(yīng)的掩碼對(duì)使得相關(guān)優(yōu)勢(shì)等于${2^{ - k}}$，給出了相關(guān)優(yōu)勢(shì)取到${2^{ - 1}}$時(shí)的充分必要條件及掩碼對(duì)的計(jì)數(shù)，給出了特定條件下非平凡相關(guān)優(yōu)勢(shì)取到最小值時(shí)的充分必要條件與掩碼對(duì)的計(jì)數(shù)。
- 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.
- SIMON algorithm /
- Linear property /
- Cyclic shift /
- S-box

HTML全文

表 1 ${F_{abc}}(x)$相關(guān)優(yōu)勢(shì)計(jì)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

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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(shí)，執(zhí)行
(7)　　$m \leftarrow \sigma \left( {\hat f,{x_v}} \right)$　　　/$\hat f$中包含${x_v}$的2次項(xiàng)個(gè)數(shù)/
(8)　　在$\hat f$中找出所有的2次項(xiàng)${x_v}{x_{{t_i}}}$滿(mǎn)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(xiàng)${x_{{t_1}}}{x_{{s_i}}}$滿(mǎn)足　　　　　　${s_1} < {s_2} < \cdots < {s_m}$，

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參考文獻(xiàn)(20)

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