SIMON64算法的積分分析

徐洪; 方玉穎; 戚文峰

doi:10.11999/JEIT190230

SIMON64算法的積分分析

doi: 10.11999/JEIT190230

徐洪^{1, 2},
方玉穎^1, ,,
戚文峰^{1, 2}

1.
信息工程大學(xué) 鄭州 450001
2.
數(shù)學(xué)工程與先進計算國家重點實驗室鄭州 450001

基金項目: 十三五國家密碼發(fā)展基金(MMJJ20180204, MMJJ20170103)

詳細(xì)信息

作者簡介:
徐洪：女，1979年生，碩士生導(dǎo)師，主要研究方向為對稱密碼的設(shè)計與分析

方玉穎：男，1994年生，碩士生，研究方向為分組密碼分析

戚文峰：男，1963 年生，教授，主要研究方向為對稱密碼的設(shè)計與分析

通訊作者:
方玉穎　fangyywy@163.com

中圖分類號: TP309.7; TN918.1
計量
- 文章訪問數(shù): 2374
- HTML全文瀏覽量: 863
- PDF下載量: 113
- 被引次數(shù): 0
出版歷程
- 收稿日期: 2019-04-09
- 修回日期: 2019-12-04
- 網(wǎng)絡(luò)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系列算法自提出以來便受到了廣泛關(guān)注。積分分析方面，Wang，F(xiàn)u和Chu等人給出了SIMON32和SIMON48算法的積分分析，該文在已有的分析結(jié)果上，進一步考慮了更長分組的SIMON64算法的積分分析?；赬iang等人找到的18輪積分區(qū)分器，該文先利用中間相遇技術(shù)和部分和技術(shù)給出了25輪SIMON64/128算法的積分分析，接著利用等價密鑰技術(shù)進一步降低了攻擊過程中需要猜測的密鑰量，并給出了26輪SIMON64/128算法的積分分析。通過進一步的分析，該文發(fā)現(xiàn)高版本的SIMON算法具有更好抵抗積分分析的能力。
- 等價密鑰 /
- SIMON64 /
- 中間相遇 /
- 部分和 /
- 積分分析
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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圖 1 SIMON算法的輪函數(shù)

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圖 2 SIMON 64算法的18輪積分區(qū)分器

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圖 3 25輪積分分析的密鑰恢復(fù)過程

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圖 4 計算${X_{18,\left\{ {31} \right\}}} \wedge {X_{18,\left\{ {24} \right\}}}$的過程

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圖 5 計算${\left( {{X_{18}} \oplus {X_{19}}} \right)_{\left\{ {30} \right\}}}$的過程

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圖 6 等價密鑰技術(shù)示意圖

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圖 7 SIMON64/128算法的26輪積分分析

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圖 8 計算$ \oplus {M_1}$的過程

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圖 9 計算$ \oplus {M_{\rm{2}}}$的過程

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表 1 計算$ \oplus \left( {{X_{18,\left\{ {31} \right\}}} \wedge {X_{18,\left\{ {24} \right\}}}} \right)$的復(fù)雜度

步驟	猜測密鑰(比特數(shù))	統(tǒng)計狀態(tài)(比特數(shù))	時間復(fù)雜度
(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 計算$ \oplus {\left( {{X_{18}} \oplus {X_{19}}} \right)_{\left\{ {30} \right\}}}$的復(fù)雜度

步驟	猜測密鑰(bit數(shù))	統(tǒng)計狀態(tài)(bit數(shù))	時間復(fù)雜度
(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 計算$ \oplus {M_1}$值的復(fù)雜度

步驟	猜測密鑰(比特數(shù))	統(tǒng)計狀態(tài)(比特數(shù))	時間復(fù)雜度
(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 計算$ \oplus {M_{\rm{2}}}$值的復(fù)雜度

步驟	猜測密鑰(比特數(shù))	統(tǒng)計狀態(tài)(比特數(shù))	時間復(fù)雜度
(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算法的積分分析(分組長度64/96/128-bit)

算法	區(qū)分器輪數(shù)	數(shù)據(jù)量(CP)	攻擊輪數(shù)	猜測密鑰量(bit)	攻擊復(fù)雜度(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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