Integral Cryptanalysis of ACE Encryption Algorithm
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摘要: ACE是国际轻量级密码算法标准化征集竞赛第2轮候选算法之一。该算法具有结构简洁，软硬件实现快、适用于资源受限环境等特点，其安全性备受业界广泛关注。该文引入字传播轨迹新概念，构建了一个传播轨迹的描述模型，并给出一个可以自动化评估分组密码算法抵抗积分攻击能力的方法。基于ACE算法结构特点，将该自动化搜索方法应用于评估ACE算法的安全性。结果表明：ACE置换存在12步的积分区分器，需要的数据复杂度为2256，时间复杂度为2256次12步的ACE置换运算，存储复杂度为8 Byte。相比于ACE算法设计者给出的积分区分器，该新区分器的步数提高了4步。Abstract: ACE as an authenticated encryption algorithm is selected as one of the round 2 candidates of the lightweight crypto standardization process. Since its excellent design advantages, e.g. simple structure, high performance in software and hardware, and suitable for constrained environments, the security of ACE is received extensive attention. In this paper, the concept of word propagation trail is introduced, and an exact model is constructed to describe the trail. A new automatic method for evaluating the security of word-based cipher against the integral attack is also proposed by using this model. Moreover, based on the structure of ACE, the security of ACE permutation is evaluated by using this new automatic method. More specifically, a new 12-step integral distinguisher of ACE permutation is verified by using this method, which requires the data complexity of about 2256 chosen data, the time complexity of about 2256 12-step ACE permutation operations, and the memory complexity of about 8 Byte. Compared with the distinguishers given by ACE’s designer, this new result prominently increases 4 steps indeed.
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表 1 文中用到的符号
符号 定义 ${{X}} \oplus {{Y}}$ 表示${{X}}$和${{Y}}$之间按位异或 +/– 十进制加/减 ${{X}}||{{Y}}$ 表示${{X}}$和${{Y}}$串联 ${1^n}/{0^n}$ 表示$n$ bit全1或全0的比特串 ${\rm{SB}}_j^i({{X}})$ 表示ACE置换第$i$步的第$j$个密码S盒 ${\rm{|}}{{X}}{\rm{|}}$ 表示集合${{X}}$中元素的个数 $(\alpha ,{{\beta }}) = \max (x[0],x[1], ··· ,x[n - 1])$ 计算数组$[x[0],x[1], ··· ,x[n - 1]]$的最大值$\alpha $以及最大值对应的下标的集合${{\beta }}$，例如$\max (2,2,1,0) = (2,[0,1])$ $\alpha = {\max '}(x[0],x[1], ··· ,x[n - 1])$ 计算数组$[x[0],x[1], ··· ,x[n - 1]]$中的最大值$\alpha $ $(\alpha ,{{\beta }}) = \min (x[0],x[1], ··· ,x[n - 1])$ 计算数组$[x[0],x[1] ··· ,x[n - 1]]$的最小值$\alpha $以及最小值对应的下标的集合${{\beta }}$，例如$\min (2,2,1,0) = (0,[3])$ $\alpha = {\min'}(x[0],x[1], ··· ,x[n - 1])$ 计算数组$[x[0],x[1], ··· ,x[n - 1]]$的最小值$\alpha $ 
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表 2 算法1：利用字传播模型确定
$r_{\rm{e}}^k$ 输入：分组密码轮函数${f_{\rm{r}}} \in {(F_2^n)^m}$，加密轮数$R$，活跃字的下标$k$ 输出：$r_{\rm{e}}^k$，以及加密$r_{\rm{e}}^k$轮后，当第$k$个字活跃时，输出状态中的平衡字的下标集合${{{\omega }}_k}$ (1) ${r_{\rm{h}}} = R$, ${r_{\rm{l}}} = 0$, $r_{\rm{e}}^k = 0$, $r = 0$, flag= 0 (2) $x_k^i[0]$, $x_k^i[1]$, $ ··· $, $x_k^i[m - 1]$为$i$轮加密后，输出状态对应的MILP模型变量，每一个MILP变量的值范围是大于等于0的整数 (3) While ${r_{\rm{h}}} - {r_{\rm{l}}} > 1$ do (4) $r = \left\lfloor {({r_{\rm{h}}} + {r_{\rm{l}}})/2} \right\rfloor $ (5) 利用性质3和性质4构建出$r$轮的字传播模型${M_{\rm{e}}}$ (6) ${M_{\rm{e}}}.{\rm{con}} \leftarrow x_k^0[k] = 1,M.{\rm{con}} \leftarrow x_k^0[j] = 0,j \in [0,m), j \ne k$ (7) ${M_{\rm{e}}}.{\rm{con}} \leftarrow {\min' }\{ (x_k^r[0],x_k^r[1], \cdots ,x_k^r[m - 1])\} = 1$ (8) 利用求解器对模型${M_{\rm{e}}}$进行求解 (9) If Me 有解 (10) ${r_{\rm{l}}} = r,{\rm{ flag }} = 1$ (11) else (12) ${r_{\rm{h}}} = r,{\rm{ flag}} = 0$ (13) End If (14) End While (15) If ${\rm{flag}} = = 1$ (16) $r_{\rm{e}}^k = r$ (17) else (18) $r_{\rm{e}}^k = r - 1$ (19)End If (20)$(1,{{{\omega }}_k}) = \min \{ (x_k^{r_{\rm{e}}^k}[0],x_k^{r_{\rm{e}}^k}[1], \cdots ,x_k^{r_{\rm{e}}^k}[m - 1])\} $ (21) return $r_{\rm{e}}^k$和${{{\omega }}_k}$
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表 3 算法2：利用字传播模型确定
$r_{\rm{d}}^k$ 输入：分组密码解密轮函数${f_{\rm{r}}}^{ - 1} \in {(F_2^n)^m}$，解密轮数$R$，活跃字的下标$k$ 输出：$r_{\rm{d}}^k$，以及解密$r_{\rm{d}}^k$轮后，输出状态中的不包含第$k$个字的下标集合${{{\varphi }}_k}$ (1) ${r_{\rm{h}}} = R$, ${r_{\rm{l}}} = 0$, $r_{\rm{d}}^k = 0$, $r = 0$, ${\rm{flag}} = 0$ (2) $y_k^i[0]$, $y_k^i[1]$, $ ··· $, $y_k^i[m - 1]$为第$i$轮解密输出状态对应的MILP模型变量，每一个MILP变量的值范围是大于等于0的整数 (3) While ${r_{\rm{h}}} - {r_{\rm{l}}} > 1$ do (4) $r = \left\lfloor {({r_{\rm{h}}} + {r_{\rm{l}}})/2} \right\rfloor $ (5) 利用性质3和性质4构建出$r$轮的字解密传播模型${M_{\rm{d}}}$ (6) ${M_{\rm{d}}}.{\rm{con}} \leftarrow y_k^0[k] = 1,M.{\rm{con}} \leftarrow y_k^0[j] = 0,j \in [0,m),j \ne k$ (7) ${M_{\rm{d}}}.{\rm{con}} \leftarrow {\min '}\{ (y_k^r[0],y_k^r[1], ··· ,y_k^r[m - 1])\} = 0$ (8) 利用求解器对模型${M_{\rm{d}}}$进行求解 (9) If Md 有解 (10) ${r_{\rm{l}}} = r,{\rm{ flag }} = 1$ (11) else (12) ${r_{\rm{h}}} = r,{\rm{ flag}} = 0$ (13) End If (14) End While (15) If ${\rm{flag}} = = 1$ (16) $r_{\rm{d}}^k = r$ (17) else (18) $r_{\rm{d}}^k = r - 1$ (19) End If (20) $(0,{{{\varphi }}_k}) = \min \{ (y_k^{r_{\rm{d}}^k}[0],y_k^{r_{\rm{d}}^k}[1], ··· ,y_k^{r_{\rm{d}}^k}[m - 1])\} $ (21) return $r_{\rm{d}}^k$和${{{\varphi }}_k}$
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表 4 ACE置换对应的
$r_{\rm{e}}^k$ 和${{{\omega }}_k}$ $k$ $r_{\rm{e}}^k$ ${{{\omega }}_k}$ 0 8 [1] 1 8 [3] 2 7 [1] 3 9 [1] 4 7 [3]
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表 5 ACE置换对应的
$r_{\rm{d}}^k$ 和${{{\varphi }}_k}$ $k$ $r_{\rm{d}}^k$ ${{{\varphi }}_k}$ 0 4 [0] 1 3 [4] 2 5 [0] 3 3 [0] 4 4 [4]
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表 6 ACE置换12步的积分区分器
输入形式 输出形式 ${c^{64}}{a^{64}}{a^{64}}{a^{64}}{a^{64}}$ ${u^{64}}{b^{64}}{u^{64}}{u^{64}}{u^{64}}$ 注：${c^{64}}$表示任意的一个64 bit的常数，${a^{64}}$表示64 bit的活跃字集，${b^{64}}$表示64 bit的平衡字集，${u^{64}}$表示64 bit的未知字集。
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