SIMON64算法的积分分析

徐洪; 方玉颖; 戚文峰

doi:10.11999/JEIT190230

SIMON64算法的积分分析

doi: 10.11999/JEIT190230

徐洪^{1, 2},
方玉颖^1, ,,
戚文峰^{1, 2}

1.
信息工程大学郑州 450001
2.
数学工程与先进计算国家重点实验室郑州 450001

基金项目: 十三五国家密码发展基金(MMJJ20180204, MMJJ20170103)

详细信息

作者简介:
徐洪：女，1979年生，硕士生导师，主要研究方向为对称密码的设计与分析

方玉颖：男，1994年生，硕士生，研究方向为分组密码分析

戚文峰：男，1963 年生，教授，主要研究方向为对称密码的设计与分析

通讯作者:
方玉颖　fangyywy@163.com

中图分类号: TP309.7; TN918.1
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- 被引次数: 0
出版历程
- 收稿日期: 2019-04-09
- 修回日期: 2019-12-04
- 网络出版日期: 2019-12-10
- 刊出日期: 2020-03-19

Integral Attacks on SIMON64

Hong XU^{1, 2},
Yuying FANG^{1
, ,},
Wenfeng QI^{1, 2}

1.
Information Engineering University, Zhengzhou 450001, China
2.
State Key Laboratory of Mathematical Engineering and Advanced Computing, Zhengzhou 450001, China

Funds: The National Cryptography Development Fund (MMJJ20180204, MMJJ20170103)

摘要

摘要:
SIMON系列算法自提出以来便受到了广泛关注。积分分析方面，Wang，Fu和Chu等人给出了SIMON32和SIMON48算法的积分分析，该文在已有的分析结果上，进一步考虑了更长分组的SIMON64算法的积分分析。基于Xiang等人找到的18轮积分区分器，该文先利用中间相遇技术和部分和技术给出了25轮SIMON64/128算法的积分分析，接着利用等价密钥技术进一步降低了攻击过程中需要猜测的密钥量，并给出了26轮SIMON64/128算法的积分分析。通过进一步的分析，该文发现高版本的SIMON算法具有更好抵抗积分分析的能力。
- 等价密钥 /
- SIMON64 /
- 中间相遇 /
- 部分和 /
- 积分分析
Abstract:
The SIMON block cipher receives extensive attention since its proposed. With respect to integral attacks, some integral attacks on SIMON32 and SIMON48 are presented by Wang, Fu and Chu et al. In this paper, on the basis of the existing analysis results, the integral attacks on SIMON64 are further studied. Based on known 18-round integral distinguisher presented by Xiang et al., the integral attacks on 25-round SIMON64/128 are presented using meet-in-the-middle and partial-sum techniques. Then the amount of subkeys that need to be guessed during the attack is further reduced by equivalent-subkey technique, and the improved integral attacks on 26-round SIMON64/128 are also presented. Through further analysis, it is found that the higher version of SIMON algorithm has better resistance to integral analysis.
- Equivalent-subkey /
- SIMON 64 /
- Meet-in-the-middle /
- Partial-sum /
- Integral attacks

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图 1 SIMON算法的轮函数

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图 2 SIMON 64算法的18轮积分区分器

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图 3 25轮积分分析的密钥恢复过程

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图 4 计算${X_{18,\left\{ {31} \right\}}} \wedge {X_{18,\left\{ {24} \right\}}}$的过程

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图 5 计算${\left( {{X_{18}} \oplus {X_{19}}} \right)_{\left\{ {30} \right\}}}$的过程

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图 6 等价密钥技术示意图

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图 7 SIMON64/128算法的26轮积分分析

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图 8 计算$ \oplus {M_1}$的过程

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图 9 计算$ \oplus {M_{\rm{2}}}$的过程

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表 1 计算$ \oplus \left( {{X_{18,\left\{ {31} \right\}}} \wedge {X_{18,\left\{ {24} \right\}}}} \right)$的复杂度

步骤	猜测密钥(比特数)	统计状态(比特数)	时间复杂度
(1)	${K_{24,\left\{ {2\sim 5,8\sim 12,14\sim 19,21\sim 26,28\sim 30} \right\}}}$(24)	${X_{24,\left\{ {4\sim 6,10\sim 13,16\sim 20,23\sim 27,30,31} \right\}}}$(19), ${Y_{24,\left\{ {2\sim 5,8\sim 12,14\sim 19,21\sim 26,28\sim 30} \right\}}}$(24)	${2^{24} } \cdot {2^{63} } \cdot \dfrac{ {24} }{ {32 \cdot 25} } \approx {2^{81.94} }$
(2)	${K_{23,\left\{ {4\sim 6,10\sim 13,16\sim 20,23\sim 27,30,31} \right\}}}$(19)	${X_{2{\rm{3}},\left\{ {0,6,7,12\sim 14,18\sim 21,25\sim 28} \right\}}}$(14), ${Y_{23,\left\{ {4\sim 6,10\sim 13,16\sim 20,23\sim 27,30,31} \right\}}}$(19)	${2^{43} } \cdot {2^{43} } \cdot \dfrac{ {19} }{ {32 \cdot 25} } \approx {2^{80.61} }$
(3)	${K_{22,\left\{ {0,6,7,12\sim 14,18\sim 21,25\sim 28} \right\}}}$(14)	${X_{2{\rm{2}},\left\{ {8,14,15,20\sim 22,27\sim 29} \right\}}}$(9), ${Y_{22,\left\{ {0,6,7,12\sim 14,18\sim 21,25\sim 28} \right\}}}$(14)	${2^{57} } \cdot {2^{33} } \cdot \dfrac{ {14} }{ {32 \cdot 25} } \approx {2^{84.16} }$
(4)	${K_{21,\left\{ {8,14,15,20\sim 22,27\sim 29} \right\}}}$(9)	${X_{2{\rm{1}},\left\{ {16,22,23,29,30} \right\}}}$(5), ${Y_{21,\left\{ {8,14,15,20\sim 22,27\sim 29} \right\}}}$(9)	${2^{66} } \cdot {2^{23} } \cdot \dfrac{9}{ {32 \cdot 25} } \approx {2^{82.53} }$
(5)	${K_{20,\left\{ {16,22,23,29,30} \right\}}}$(5)	${X_{{\rm{20,}}\left\{ {{\rm{24,31}}} \right\}}}$(2), ${Y_{20,\left\{ {16,22,23,29,30} \right\}}}$(5)	${2^{71} } \cdot {2^{14} } \cdot \dfrac{5}{ {32 \cdot 25} } \approx {2^{77.68} }$
(6)	${K_{{\rm{19,}}\left\{ {{\rm{24,31}}} \right\}}}$(2)	${X_{18,\left\{ {24,31} \right\}}}$(2), ${X_{18,\left\{ {24} \right\}}} \wedge {X_{18,\left\{ {31} \right\}}}$(1)	${2^{73} } \cdot {2^7} \cdot \dfrac{3}{ {32 \cdot 25} } \approx {2^{71.95} }$

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表 2 计算$ \oplus {\left( {{X_{18}} \oplus {X_{19}}} \right)_{\left\{ {30} \right\}}}$的复杂度

步骤	猜测密钥(bit数)	统计状态(bit数)	时间复杂度
(1)	${K_{24,\left\{ {0,2\sim 4,6\sim 29} \right\}}}$(28)	${X_{2{\rm{4}},\left\{ {4,5,8,10\sim 12,14\sim 30} \right\}}}$(23), ${Y_{24,\left\{ {0,2\sim 4,6\sim 29} \right\}}}$(28)	${2^{28} } \cdot {2^{63} } \cdot \dfrac{ {28} }{ {32 \cdot 25} } \approx {2^{86.17} }$
(2)	${K_{23,\left\{ {4,5,8,10\sim 12,14\sim 30} \right\}}}$(23)	${X_{2{\rm{3}},\left\{ {6,12,13,16,18\sim 20,22\sim 30} \right\}}}$(16), ${Y_{23,\left\{ {4,5,8,10\sim 12,14\sim 30} \right\}}}$(23)	${2^{51} } \cdot {2^{51} } \cdot \dfrac{ {23} }{ {32 \cdot 25} } \approx {2^{96.88} }$
(3)	${K_{22,\left\{ {6,12,13,16,18\sim 20,22\sim 30} \right\}}}$(16)	${X_{2{\rm{2}},\left\{ {14,20,21,24,26\sim 28,30,31} \right\}}}$(9), ${Y_{22,\left\{ {6,12,13,16,18\sim 20,22\sim 30} \right\}}}$(16)	${2^{67} } \cdot {2^{39} } \cdot \dfrac{ {16} }{ {32 \cdot 25} } \approx {2^{100.36} }$
(4)	${K_{21,\left\{ {14,20,21,24,26\sim 28,30,31} \right\}}}$(9)	${X_{2{\rm{1}},\left\{ {0,22,28,29} \right\}}}$(4), ${Y_{21,\left\{ {14,20,21,24,26\sim 28,30,31} \right\}}}$(9)	${2^{76} } \cdot {2^{25} } \cdot \dfrac{9}{ {32 \cdot 25} } \approx {2^{94.53} }$
(5)	${K_{20,\left\{ {0,22,28,29} \right\}}}$(4)	${X_{{\rm{20}},\left\{ {30} \right\}}}$(1), ${Y_{20,\left\{ {0,22,28,29} \right\}}}$(4)	${2^{80} } \cdot {2^{13} } \cdot \dfrac{4}{ {32 \cdot 25} } \approx {2^{85.36} }$
(6)	${K_{19,\left\{ {30} \right\}}}$(1)	${X_{{\rm{18,}}\left\{ {{\rm{31}}} \right\}}}$(1), $ \oplus {\left( {{X_{18}} \oplus {X_{19}}} \right)_{\left\{ {30} \right\}}}$ (1)	${2^{81} } \cdot {2^5} \cdot \dfrac{2}{ {32 \cdot 25} } \approx {2^{77.36} }$

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表 3 计算$ \oplus {M_1}$值的复杂度

步骤	猜测密钥(比特数)	统计状态(比特数)	时间复杂度
(1)	$ - $	${X_{25,\left\{ {2\sim 5,8\sim 12,14\sim 19,21\sim 26,28\sim 30} \right\}}}$(24), ${Y_{25,\left\{ {0\sim 4,6\sim 29} \right\}}}$(29)	${2^{63} } \cdot \dfrac{ {29} }{ {32 \cdot 26} } \approx {2^{58.16} }$
(2)	$K_{25,\left\{ {0\sim 4,6\sim 11,13\sim 18,20\sim 29} \right\}}^{\rm{*}}$(27)	${X_{{\rm{24,}}\left\{ {{\rm{4\sim 6,10\sim 13,16\sim 20,23\sim 27,30,31}}} \right\}}}$(19), ${Y_{2{\rm{4}},\left\{ {2\sim 5,8\sim 12,14\sim 19,21\sim 26,28\sim 30} \right\}}}$(24)	${2^{27} } \cdot {2^{53} } \cdot \dfrac{ {24} }{ {32 \cdot 26} } \approx {2^{74.89} }$
(3)	$K_{{\rm{24,}}\left\{ {{\rm{2,5,9,12,16,19,23,26,30}}} \right\}}^{\rm{*}}$(9)	${X_{23,\left\{ {2,3,9,10,16,17,23,24,28} \right\}}}$(9), ${Y_{23,\left\{ {6,10,13,17,20,24,27,31} \right\}}}$(8), ${X_{24,\left\{ {4,11,18,25} \right\}}}$(4)	${2^{36} } \cdot {2^{43} } \cdot \dfrac{8}{ {32 \cdot 26} } \approx {2^{72.30} }$
(4)	$K_{24,\left\{ {3,10,17,24,28} \right\}}^*$(5)	${X_{23,\left\{ {0,3,4,6\sim 8,10\sim 15,17\sim 22,25\sim 29} \right\}}}$(23), ${Y_{23,\left\{ {4,11,18,25} \right\}}}$(4), ${X_{24,\left\{ {2,12,16,19,23,26,30} \right\}}}$(7)	${2^{41} } \cdot {2^{21} } \cdot \dfrac{4}{ {32 \cdot 26} } \approx {2^{54.30} }$
(5)	$K_{24,\left\{ {4,8,11,15,18,22,25,29} \right\}}^*$(8)	${X_{23,\left\{ {{\rm{0,6,}}7,12\sim 14,18\sim 21,25\sim 28} \right\}}}$(14), ${Y_{{\rm{23,}}\left\{ {{\rm{4\sim 6,10\sim 13,16\sim 20,23\sim 27,30,31}}} \right\}}}$(19)	${2^{49} } \cdot {2^{34} } \cdot \dfrac{ {19} }{ {32 \cdot 26} } \approx {2^{77.55} }$
(6)	$K_{23,\left\{ {4,11,18,25} \right\}}^*$(4)	${X_{22,\left\{ {4,5,11,12,18,19,25,26,30} \right\}}}$(9), ${Y_{{\rm{22,}}\left\{ {{\rm{12,19,26}}} \right\}}}$(3), ${X_{{\rm{23,}}\left\{ {{\rm{6,13,20,27}}} \right\}}}$(4)	${2^{53} } \cdot {2^{33} } \cdot \dfrac{3}{ {32 \cdot 26} } \approx {2^{77.88} }$
(7)	$K_{23,\left\{ {5,12,19,26} \right\}}^*$(4)	${X_{22,\left\{ {5,6,8,10,12\sim 17,19\sim 24,26\sim 31} \right\}}}$(22), ${Y_{22,\left\{ {6,13,20,27} \right\}}}$(4), ${X_{23,\left\{ {0,7,14,18,21,25,28} \right\}}}$(7)	${2^{57} } \cdot {2^{16} } \cdot \dfrac{4}{ {32 \cdot 26} } \approx {2^{65.30} }$
(8)	$K_{23,\left\{ {6,10,13,17,20,24,27,31} \right\}}^*$(8)	${X_{22,\left\{ {8,14,15,20\sim 22,27\sim 29} \right\}}}$(9), ${Y_{22,\left\{ {0,6,7,12\sim 14,18\sim 21,25\sim 28} \right\}}}$(14)	${2^{65} } \cdot {2^{33} } \cdot \dfrac{ {14} }{ {32 \cdot 26} } \approx {2^{92.11} }$
(9)	$K_{22,\left\{ {0,7,14,21,28} \right\}}^*$(5)	${X_{21,\left\{ {6,12,13,19,20,27,28} \right\}}}$(7), ${Y_{21,\left\{ {8,15,22,29} \right\}}}$(4), ${X_{22,\left\{ {14,21,28} \right\}}}$(3)	${2^{70} } \cdot {2^{23} } \cdot \dfrac{4}{ {32 \cdot 26} } \approx {2^{85.30} }$
(10)	$K_{22,\left\{ {6,13,20,27} \right\}}^*$(4)	${X_{21,\left\{ {12,16,18,19,22,23,25,26,29,30} \right\}}}$(10), ${Y_{21,\left\{ {14,21,28} \right\}}}$(3), ${X_{22,\left\{ {20,27} \right\}}}$(2)	${2^{74} } \cdot {2^{14} } \cdot \dfrac{3}{ {32 \cdot 26} } \approx {2^{79.88} }$
(11)	$K_{22,\left\{ {12,19,26} \right\}}^*$(4)	${X_{21,\left\{ {16,22,23,29,30} \right\}}}$(5), ${Y_{21,\left\{ {8,14,15,20\sim 22,27\sim 29} \right\}}}$(9)	${2^{77} } \cdot {2^{15} } \cdot \dfrac{9}{ {32 \cdot 26} } \approx {2^{85.47} }$
(12)	$K_{21,\left\{ {8,15,22,29} \right\}}^*$(4)	${X_{20,\left\{ {14,20,21,24,27,28,31} \right\}}}$(7), ${Y_{20,\left\{ {16,23,30} \right\}}}$(3), ${X_{21,\left\{ {22,29} \right\}}}$(2)	${2^{81} } \cdot {2^{14} } \cdot \dfrac{3}{ {32 \cdot 26} } \approx {2^{86.88} }$
(13)	$K_{21,\left\{ {14,21,28} \right\}}^*$(3)	${X_{20,\left\{ {24,31} \right\}}}$(2), ${Y_{20,\left\{ {16,22,23,29,30} \right\}}}$(5)	${2^{84} } \cdot {2^{12} } \cdot \dfrac{5}{ {32 \cdot 26} } \approx {2^{88.62} }$
(14)	$K_{20,\left\{ {16,23,30} \right\}}^*$(3)	${X_{19,\left\{ 0 \right\}}}$(1), ${Y_{19,\left\{ {24,31} \right\}}}$(2)	${2^{ {\rm{87} } } } \cdot { {\rm{2} }^{\rm{7} } } \cdot \dfrac{ {\rm{2} } }{ {32 \cdot 26} } \approx {2^{ {\rm{85} }{\rm{.30} } } }$
(15)	$K_{19,\left\{ {24,31} \right\}}^*$(2)	$\left( {{X_{18,\left\{ {31} \right\}}} \wedge {X_{18,\left\{ {{\rm{24}}} \right\}}}} \right) \oplus {X_{19,\left\{ 0 \right\}}}$(1)	${2^{ {\rm{89} } } } \cdot { {\rm{2} }^{\rm{3} } } \cdot \dfrac{ {\rm{1} } }{ {32 \cdot 26} } \approx {2^{ {\rm{82} }{\rm{.30} } } }$

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表 4 计算$ \oplus {M_{\rm{2}}}$值的复杂度

步骤	猜测密钥(比特数)	统计状态(比特数)	时间复杂度
(1)	$ - $	${X_{25,\left\{ {2\sim 4,8\sim 11,14\sim 18,20\sim 24,28,29} \right\}}}$(19), ${Y_{25,\left\{ {0\sim 3,6\sim 10,12\sim 23,26\sim 28} \right\}}}$(24)	${2^{63} } \cdot \dfrac{ { {\rm{24} } } }{ {32 \cdot 26} } \approx {2^{5{\rm{7} }{\rm{.89} } } }$
(2)	$K_{25\left\{ {0\sim 3,6\sim 10,12\sim 17,19\sim 23,26\sim 28} \right\}}^*$(22)	${X_{24,\left\{ {4,5,10\sim 12,16\sim 19,22\sim 25,30} \right\}}}$(14), ${Y_{24,\left\{ {2\sim 4,8\sim 11,14\sim 18,20\sim 24,28,29} \right\}}}$(19)	${2^{ {\rm{22} } } } \cdot { {\rm{2} }^{ {\rm{43} } } } \cdot \dfrac{ { {\rm{19} } } }{ {32 \cdot 26} } \approx {2^{ {\rm{59} }{\rm{.55} } } }$
(3)	$K_{24\left\{ {2\sim 4,8\sim 11,14\sim 18,21\sim 24,28,29} \right\}}^*$(18)	${X_{23,\left\{ {6,12,13,18\sim 20,24\sim 27} \right\}}}$(10), ${Y_{23,\left\{ {4,5,10\sim 12,16\sim 19,22\sim 25,30} \right\}}}$(14)	${2^{ {\rm{40} } } } \cdot { {\rm{2} }^{ {\rm{33} } } } \cdot \dfrac{ { {\rm{14} } } }{ {32 \cdot 26} } \approx {2^{ {\rm{67} }{\rm{.11} } } }$
(4)	$K_{23,\left\{ {4,5,10\sim 12,16\sim 19,23\sim 25,30} \right\}}^*$(13)	${X_{22,\left\{ {14,20,21,26\sim 28} \right\}}}$(6), ${Y_{22,\left\{ {6,12,13,18\sim 20,24\sim 27} \right\}}}$(10)	${2^{ {\rm{5} }3} } \cdot { {\rm{2} }^{ {\rm{24} } } } \cdot \dfrac{ { {\rm{1} }0} }{ {32 \cdot 26} } \approx {2^{ {\rm{70} }{\rm{.63} } } }$
(5)	$K_{22\left\{ {6,12,13,18\sim 20,25\sim 27} \right\}}^*$(9)	${X_{21,\left\{ {22,28,29} \right\}}}$(3), ${Y_{21,\left\{ {14,20,21,26\sim 28} \right\}}}$(6)	${2^{6{\rm{2} } } } \cdot { {\rm{2} }^{ {\rm{16} } } } \cdot \dfrac{ {\rm{6} } }{ {32 \cdot 26} } \approx {2^{ {\rm{70} }{\rm{.89} } } }$
(6)	$K_{21\left\{ {14,20,21,27,28} \right\}}^*$(5)	${X_{20,\left\{ {30} \right\}}}$(1), ${Y_{20,\left\{ {22,28,29} \right\}}}$(3)	${2^{6{\rm{7} } } } \cdot { {\rm{2} }^{\rm{9} } } \cdot \dfrac{3}{ {32 \cdot 26} } \approx {2^{ {\rm{67} }{\rm{.89} } } }$
(6)	$K_{20\left\{ {22,29} \right\}}^*$(2)	$ \oplus {Y_{19,\left\{ {30} \right\}}}$(1)	${2^{6{\rm{9} } } } \cdot { {\rm{2} }^{\rm{4} } } \cdot \dfrac{ {\rm{1} } }{ {32 \cdot 26} } \approx {2^{ {\rm{63} }{\rm{.30} } } } $

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表 5 SIMON算法的积分分析(分组长度64/96/128-bit)

算法	区分器轮数	数据量(CP)	攻击轮数	猜测密钥量(bit)	攻击复杂度(E)
SIMON64/96	18	${{\rm{2}}^{63}}$	25	73	${{\rm{2}}^{{\rm{95}}}}$
SIMON64/128	18	${{\rm{2}}^{63}}$	26	102	${{\rm{2}}^{127}}$
SIMON96/96	22	${{\rm{2}}^{{\rm{95}}}}$	28	64	${{\rm{2}}^{{\rm{95}}}}$
SIMON96/144	22	${{\rm{2}}^{{\rm{95}}}}$	30	138	${{\rm{2}}^{{\rm{95}}}}$
SIMON128/128	26	${{\rm{2}}^{{\rm{127}}}}$	33	98	${{\rm{2}}^{{\rm{127}}}}$
SIMON128/192	26	${{\rm{2}}^{{\rm{127}}}}$	35	187	${{\rm{2}}^{{\rm{127}}}}$
SIMON128/256	26	${{\rm{2}}^{{\rm{127}}}}$	36	241	${{\rm{2}}^{{\rm{127}}}}$

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