A Lattice Cipher Template Attack Method Based on Recurrent Cryptography
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摘要: 该文分析了格密码解封装过程存在的能量泄露，针对消息解码操作提出一种基于模板与密文循环特性的消息恢复方法，该方法采用汉明重量模型与归一化类间方差(NICV)方法对解码字节的中间更新状态构建模板，并利用密文循环特性构造特殊密文，结合算法运算过程中产生的能量泄露，实现了对格密码中秘密消息和共享密钥的恢复。该文以Saber算法及其变体为例对提出的攻击方法在ChipWhisperer平台上进行了验证，结果表明，该攻击方法可以成功还原封装阶段的秘密消息和共享密钥，在预处理阶段仅需900条能量迹即可完成对模板的构建，共需要32条能量迹完成秘密消息的恢复。在未增加信噪比(SNR)条件下，消息恢复成功率达到66.7%，而在合适信噪比条件下，消息恢复成功率达到98.43%。Abstract: The energy leakage in the decapsulation process of lattice-based cryptography is analyzed and a message recovery method targeting the message decoding with profiling and ciphertexts rotation is proposed in this paper. The templates are constructed using Hamming weight model as well as Normalized Inter-Class Variance (NICV) for the intermediate state of decoded bytes. The special ciphertexts are built by rotating the original ciphertexts. Combining the energy leakage generated during the calculations, the secret messages and shared keys are recovered. Experiments and tests are carried out with Saber and its variants on the ChipWhisperer-STM32F303 board and the results indicate that the proposed method can successfully recover the secret message and shared key of the encapsulation stage. It only needs 900 energy traces to complete the construction for templates and a total of 32 power traces in recovering the secret message. The success rate of message recovery reaches 66.7% under the condition of no increasing the Signal-to-Noise Ratio (SNR), and 98.43% under the condition of sufficient SNR.
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Key words:
- Lattice-based cryptography /
- Energy leakage /
- Template analysis /
- Ciphertexts rotation /
- Saber algorithm
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算法1 Saber PKE解密算法 输入：私钥$ s \in {B^*} $ 密文$c = ({c_m},{\boldsymbol{b}}') \in {B^{\text{*} } }$ 输出：解密消息$ m' \in {B^{32}} $ (1) $v = {\boldsymbol{b}}{'^{\rm{T} } }(s\text{mod} p) \in {R_p}$ (2) $ m' = ((v + {h_2} - {2^{{\varepsilon _p} - {\varepsilon _T}}}{c_m})\text{mod} p) > > ({\varepsilon _p} - 1) \in {R_2} $ (3) 返回$ m' $ 
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算法2 Saber KEM解封装算法 输入：私钥${{\rm{sk}}_{{\rm{KEM}}} } = (z,{\rm{pkh,pk}},s) \in {B^{\text{*} } }$ 密文$ c \in {B^{\text{*}}} $ 输出：共享密钥$ K \in {B^{32}} $ (1) $m' = {\rm{Saber}}\; {\rm{PKE.Dec}}(s,c)$ (2) $(K',r') = G({\rm{pkh}},m')$ (3) $c' = {\rm{Saber} }\; {\rm{PKE.Enc} }({\rm{pk}},m';r')$ (4) 若 $ c = c' $ 则返回$K = {\rm{KDF}}(K'||H(c))$ (5) 否则返回$K = {\rm{KDF}}(z||H(c))$
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算法3 Saber POLmsg2BS (1) void POLmsg2BS(uint8_t bytes[SABER_KEYBYTES],
const poly *data) {(2) size_t t, i, j; (3) for (j = 0; j < SABER_KEYBYTES; j++) { (4) bytes[j] = 0; (5) for (i = 0; i < 8; i++) { (6) t = (data->coeffs[j × 8 + i] & 0x01); (7) bytes[j] |= t << i; (8) } (9) } (10) }
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算法4 针对Saber算法的消息恢复方法 预处理阶段 (1) for j from 0 by 1 to 7 do (2) for k from 0 by 1 to j+1 do (3) $T_{\left( {0,j} \right)}^k = {{\rm{Saber}}} {{\rm{KEM}}} .{\rm{Dec}}({\rm{CT}}_{\left( {0,j} \right)}^k)$ (4) end for (5) ${W_j} = {{\rm{TVLA}}} (T_{(0,1)}^0,T_{\left( {0,1} \right)}^1)$ (6) ${P_{(0,j)} } = {\rm{NICV}}({W_j},T_{\left( {0,j} \right)}^0,\cdots,T_{\left( {0,j} \right)}^{j + 1})$ (7) for k from 0 by 1 to j+1 do (8) $T{_{\left( {0,j} \right)}^{k'}} = T_{\left( {0,j} \right)}^k({P_{(0,j)} })$ (9) $r_P^k = {\text{average} }(T{_{\left( {0,j} \right)}^{k'}})$ (10) end for (11) end for 模板匹配阶段 (12) for i from 0 by 1 to l –1 do (13) cti = 构造密文(ct, i) (14) tri =Saber KEM.Dec(cti) (15) for j from 0 by 1 to 7 do (16) ${ {\rm{tr} }'_{(i,j)} } = { {\rm{tr} }_i}({P_{(0,j)} })$ (17) for k from 0 by 1 to j+1 do (18) $\varGamma _{(i,j)}^k = { {\rm{SOSD} } } ({ {\rm{tr} }'_{(i,j)} },r_P^k)$ (19) end for (20) ${{\rm{HW}}} (m[i,j]) = \min (\varGamma _{(i,j)}^k)$ (21) $m{[i]_j} = {\text{Recover} }({{\rm{HW}}} (m[i,j]),{{\rm{HW}}} (m[i,j] - 1))$ (22) end for (23) end for
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表 2 不同SNR条件下恢复秘密消息的成功率
实验 重复测量次数 成功率*(%) 实验1 1 67.50/66.71/66.68 实验2 2 90.00/89.67/89.63 实验3 4 95.81/95.64/95.64 实验4 6 97.36/97.22/97.20 实验5 8 98.62/98.43/98.37 实验6 10 98.62/98.43/98.39 实验7 12 98.62/98.43/98.39 *：成功率分别为LightSaber/Saber/FireSaber
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