Integral Cryptanalysis and Impossible Differential Cryptanalysis of the μ2 Algorithm
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摘要:
$ {\mu }^{\text{2}} $ 算法是由Yeoh等人设计的一种轻量级分组密码算法(doi:10.1007/978-981-15-0058-9-27 )，该算法全轮共15轮，采用TYPE-II广义Feistel结构，Yeoh等人在设计文档中对$ {\mu ^{\text{2}}} $ 算法抵抗差分分析、线性分析的能力进行了评估，但$ {\mu ^{\text{2}}} $ 算法抵抗积分攻击和不可能差分分析的能力目前尚不清楚。该文给出了$ {\mu ^{\text{2}}} $ 算法的8轮和9轮积分区分器和9轮不可能差分，利用8轮积分区分器，对9轮$ {\mu ^{\text{2}}} $ 算法进行了积分攻击，攻击的时间复杂度为${2^{76}}$ 次9轮加密，数据复杂度为${2^{48}}$ ，存储复杂度为${2^{48}}$ ；利用9轮不可能差分，对11轮$ {\mu ^{\text{2}}} $ 算法进行了不可能差分分析，攻击的时间复杂度为${2^{49}}$ 次11轮加密，数据复杂度为${2^{64}}$ 对明文。结果表明，9轮的$ {\mu ^{\text{2}}} $ 算法不能抵抗积分攻击，11轮的$ {\mu ^{\text{2}}} $ 算法不能抵抗不可能差分分析。另外，该文对$ {\mu ^{\text{2}}} $ 算法抵抗差分攻击的能力进一步评估并证明4轮$ {\mu ^{\text{2}}} $ 算法的差分特征的最大概率为${{\text{2}}^{{{ - 39}}}}$ ，与设计报告指出的4轮差分特征的概率不超过${2^{ - 3{\text{6}}}}$ 相比结果更为紧致。-
关键词:
- $ {\mu ^{\text{2}}} $算法 /
- 混合整数线性规划 /
- 积分分析 /
- 不可能差分分析 /
- 差分分析
Abstract:$ {\mu ^{\text{2}}} $ is a lightweight block cipher designed by Yeoh et al (doi:10.1007/978-981-15-0058-9-27 ). The cipher has 15 rounds in total and adopts TYPE-II generalized feistel network. The ability of the$ {\mu ^{\text{2}}} $ algorithm to resist differential analysis and linear analysis is evaluated by Yeoh et al. in the design document, but the ability of$ {\mu ^{\text{2}}} $ algorithm to resist integral attack and impossible differential attack is not clear. In this paper, 8/9-round integral distinguishers and 9-round impossible difference are given. Using 8-round integral distinguishers, 9-round$ {\mu ^{\text{2}}} $ is attacked and the time complexity of the attack is${2^{76}}$ 9-round encryptions, the data complexity is${2^{48}}$ , and the memory complexity is${2^{48}}$ . Using 9-round impossible difference, 11-round$ {\mu ^{\text{2}}} $ is attacked and the time complexity of the attack is${2^{49}}$ 11-round encryptions, the data complexity is${2^{64}}$ pairs of plaintexts. The results show that the 9-round$ {\mu ^{\text{2}}} $ algorithm can not resist integral attack, and the 11-round$ {\mu ^{\text{2}}} $ algorithm can not resist impossible differential analysis. In addition, the ability of$ {\mu ^{\text{2}}} $ algorithm to resist differential attack is reevaluated, and the maximum probability of differential characteristic of 4-round$ {\mu ^{\text{2}}} $ algorithm is${{\text{2}}^{{{ - 39}}}}$ , which is more compact than the probability${2^{ - 3{\text{6}}}}$ of 4-round differential characteristic pointed out in the design report. -
表 2 积分区分器
轮数 区分器 8轮 输入 cccccccccccccccc, aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaaa 输出 ????????????????, ????????????????, bbbbbbbbbbbbbbbb, ???????????????? 输入 aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaaa, cccccccccccccccc, aaaaaaaaaaaaaaaa 输出 bbbbbbbbbbbbbbbb, ????????????????, ????????????????, ???????????????? 9轮 输入 aaaaaaaaaaaaaaac, aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaaa 输出 bbbbbbbbbbbbbbbb, ????????????????, ????????????????, ???????????????? 输入 aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaaa, aaaaaaaaaaaaaaac, aaaaaaaaaaaaaaaa 输出 ????????????????, ????????????????, bbbbbbbbbbbbbbbb, ???????????????? 
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表 3 4轮
${\mu ^2}$ 算法最优差分特征轮数 输入差分 第1轮输入 0000000000000000 0000000010110111 1010010100000010 0010000100000101 第2轮输入 0000000010110111 1010010100000000 0000000000000000 0000000000000000 第3轮输入 0000000000000000 0000000000000000 0000000000000000 0000000010110111 第4轮输入 0000000000000000 0000000000000000 0000000010110111 0000000000000000 第4轮输出 0000000000000000 0000000000000000 0000000010110111 1000010100000001
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