基于动态补偿的椭圆曲线密码低成本抗功耗攻击策略及硬件结构研究

李伟; 曾涵; 陈韬; 南龙梅

doi:10.11999/JEIT210581

基于动态补偿的椭圆曲线密码低成本抗功耗攻击策略及硬件结构研究

doi: 10.11999/JEIT210581

李伟^1, ,,
曾涵¹,
陈韬¹,
南龙梅²

1.
解放军信息工程大学郑州 450000
2.
复旦大学专用集成电路与系统国家重点实验室上海 200433

基金项目: 国家科技重大专项(2018ZX01027101-004)，基础加强计划基金(2019-JCJQ-JJ-123)

详细信息

作者简介:
李伟：男，1983年生，副教授，博士生导师，研究方向为密码处理器设计，ASIC专用芯片设计

曾涵：女，1998年生，硕士生，研究方向为安全SoC与专用指令处理器设计

陈韬：男，1979年生，副教授，硕士生导师，研究方向为安全专用芯片设计

南龙梅：女，1981年生，博士生，研究方向为大规模集成电路设计、专用集成电路设计

通讯作者:
李伟　liwei12@fudan.edu.cn

中图分类号: TN918.2; TP316.4
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出版历程
- 收稿日期: 2021-06-16
- 修回日期: 2021-08-16
- 网络出版日期: 2021-08-27
- 刊出日期: 2021-09-16

Dynamic Compensation Based Low-cost Power-analysis Countermeasure for Elliptic Curve Cryptography and Its Hardware Structure

Wei LI^{1
, ,},
Han ZENG¹,
Tao CHEN¹,
Longmei NAN²

1.
PLA Information Engineering University, Zhengzhou 450000, China
2.
State Key Laboratory of ASIC and System, Fudan University, Shanghai 200433, China

Funds: The National Science and Technology Major Project (2018ZX01027101-004), The Foundation Strengthening Program (2019-JCJQ-JJ-123)

摘要

摘要: 椭圆曲线密码(ECC)芯片的抗功耗攻击能力往往以电路性能、面积或功耗为代价。该文分析了在椭圆曲线密码点乘运算中密钥猜测正确与错误时的中间数据汉明距离概率分布差异性，提出一种基于动态汉明距离调控的功耗补偿方法，利用模拟退火算法离线寻找最优的映射矩阵，最终形成椭圆曲线密码硬件电路的等概率映射补偿模型，大大降低了中间数据与功耗的相关性。同时，以该模型为指导设计了低成本的同步功耗补偿电路，在CMOS 40 nm工艺下，防护后的ECC128电路面积增加22.8%。基于Sakura-G开发板开展了测试验证，防护电路的功耗仅增加18.8%，最小泄露轨迹数大于10⁴，抗相关功耗分析能力提升了312倍。该策略在与随机化方法防护能力相当的情况下，不损失电路性能且硬件成本小，适用于高速或资源受限的ECC电路。
- 椭圆曲线密码 /
- 相关功耗分析 /
- 低成本 /
- 模拟退火算法
Abstract: The power-analysis countermeasure for Elliptic Curve Cryptographic (ECC) chips endures large area, power consumption and performance degradation. In this paper, the difference in the probability distribution of the intermediate data Hamming distance is analyzed when the key guess is correct and incorrect in the point multiplication of ECC. A power compensation method based on dynamic Hamming distance control is proposed, which uses the simulated annealing algorithm offline to find the optimal mapping matrix. Finally, a mapping compensation model of equal probability on the elliptic curve cryptographic hardware is formed, which greatly reduces the correlation between intermediate data and power consumption. At the same time, a low-cost synchronous power compensation circuit is designed in the guidance of this model. Under the CMOS 40 nm process, the area of protected ECC128 is only increased by 22.8%. Experiments and tests are carried out on the Sakura-G board. The power overhead is 18.8%, and the number of minimum leakage traces is greater than 10⁴, which is increased by 312 times. This countermeasure is the same as randomization with low cost and no impact on the throughput rate, which is suitable for high-speed or resource-constrained ECC circuits.
- Elliptic Curve Cryptography (ECC) /
- Correlation Power Analysis (CPA) /
- Low cost /
- Simulated annealing algorithm

HTML全文

图 1 椭圆曲线密码各层次运算

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图 2 正态分布对模运算的中间值汉明距离拟合程度

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图 3 不同有限域P下的中间数据汉明距离的均值与标准

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图 4 不同参数h下，针对ECC128的模拟退火算法收敛时间与补偿后的平均汉明距离

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图 5 转换电路结构图

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图 6 补偿电路的硬件实现与时序逻辑

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图 7 基于Sakura-G开发板的安全测试平台

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图 8 CPA攻击相关系数与最小泄露轨迹数

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表 1 Radix-4交错模乘算法

输入：A, B, P, 位宽m 　输出：$V = A \cdot B\boldsymbolod P$
(1) $V = {b_0} \cdot A$, $U = 2A\boldsymbolod P$, $A = 4A\boldsymbolod P$, $B = B/2$；　(2) For i from 0 to $\left\lceil { {{m} }/2} \right\rceil - 1$ 　$V = (V + {b_0} \cdot U + {b_1} \cdot A)\boldsymbolod P$, $U = 2A\boldsymbolod P$, 　$A = 4A\boldsymbolod P$, $B{\rm{ = }}B/4$；　(3) Return V

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表 2 参数列表

参数	含义
num	模乘除运算中间数据的概率分布函数之差的个数
${f_{{\rm{wsubr(}}j{\rm{)}}}}$	第j个密钥猜测错误与正确时的离散概率分布函数之差
mmd	ECC硬件实现时，在能量迹上显示出的模乘和模除数量之和
n	ECC硬件实现中，中间数据汉明距离的所有可能汉明距离的个数
h	衡量等概率映射时的概率参数，汉明距离以概率2^–^h被映射为2^h个值
M	为一个n行2^h的概率矩阵，矩阵中的第x行元素$({m_{x,1} },{m_{x,2} }, ··· ,{m_{x,{2^{{h} } } }})$表示将实际汉明距离x分别以相等的概率2^-^h映射为${m_{x,1} },{m_{x,2} }, ··· ,{m_{x,{2^{{h} } } }}$
M_i	矩阵的第i列向量

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表 3 寻找最优M矩阵的模拟退火算法

　输入：代价函数${\rm{cos}}{{\rm{t}}_{{\rm{SUM}}}}$，降温系数α，代价函数阈值${\rm{threshold}}$，矩阵维度n, h；
　输出：映射矩阵${\boldsymbol{M'}}$。

　(1) 初始化${\boldsymbol{M}}$矩阵元素${m_{i,j}} = i$，温度Tmp；
　(2) 计算代价函数${\rm{cos}}{{\rm{t}}_{{\rm{old}}}} = {\rm{cos}}{{\rm{t}}_{{\rm{SUM}}}}({\boldsymbol{M}})$；
　(3) 生成随机向量${\boldsymbol{r} } = {({r_1},{r_2}, ···, {r_n})_n}$，其中${r_{\rm{i}}}{\rm{ = (}}i{\rm{ - 1) + (}}n{\rm{ - }}i{\rm{ + 1)}} \cdot {\rm{ran}}{{\rm{d}}_i}$, ${\rm{ran}}{{\rm{d}}_i}$为0～1之间的随机数；将矩阵的一列向量更新为
　　 ${ {\boldsymbol{M} }_i} = \left\lceil {\boldsymbol{r} } \right\rceil = {({\left\lceil r \right\rceil _1},\left\lceil { {r_2} } \right\rceil , ···, \left\lceil { {r_n} } \right\rceil )^{\rm{T} } }$；
　(4) 重复步骤 3，直到矩阵的h个列向量全部被替换，生成新的${{\boldsymbol{M}}_{{\rm{new}}}}$；
　(5) 计算新的代价函数${\rm{cos}}{{\rm{t}}_{{\rm{new}}}}$，以及$\delta = {\rm{cos}}{{\rm{t}}_{{\rm{new}}}} - {\rm{cos}}{{\rm{t}}_{{\rm{old}}}}$；
　(6) 生成一个0～1的随机数R，若$\delta < 0$，则${\boldsymbol{M}} = {{\boldsymbol{M}}_{{\rm{new}}}}$, ${\rm{cos}}{{\rm{t}}_{{\rm{old}}}}{\rm{ = cos}}{{\rm{t}}_{{\rm{new}}}}$；
　否则，若$\exp ({\rm{ - }}\delta /{\rm{Tmp}}) > R$，则${\boldsymbol{M}} = {{\boldsymbol{M}}_{{\rm{new}}}}$, ${\rm{cos}}{{\rm{t}}_{{\rm{old}}}}{\rm{ = cos}}{{\rm{t}}_{{\rm{new}}}}$，并进行降温，令${\rm{Tmp}} = {\rm{Tmp}} \cdot \alpha $；
　(7) 若${\rm{cos}}{{\rm{t}}_{{\rm{old}}}}{\rm{ > threshold}}$，则返回步骤 3；
　(8) 令${\boldsymbol{M}}$中的元素${m_{i,j}} = {m_{i,j}} - i$，生成${\boldsymbol{M'}}$矩阵并返回。

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表 4 由${c_i}$输出值推导出的${a_i}$真值表

${ {{p} }_i}$	${c_i}$	${c_{i{\rm{ - }}1}}$	${a_i}$	${a'_i}$
0	0	0	0	0
0	0	0	1	0
0	0	1	0	0
0	1	1	1	1
0	1	0	–	X
1	0	0	0	0
1	1	0	1	1
1	1	1	0	0
0	1	1	1	0
1	0	1	–	X

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表 5 防护前后代价与性能对比

	综合面积(mm²/kGates)	关键路径延时(ns)	FPGA LUTs/FFs	FPGA 功耗(W)
防护前	0.281/293.4	3.21	15851/3574	0.101
防护后	0.345/360.3	3.21	19687/4285	0.120
增加的百分比	+22.8%	0	+24.2%/+19.9%	+18.8%

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表 6 防护能力与电路代价对比分析

	VLSI’14 ^[6]	TIE’16^[7]	TDSC’18^[16]	TCAS-I’21^[17]	本文
原理	基点随机化	中间点随机化	标量k随机化	射影坐标随机化	功耗补偿
域-位宽(bit)	素域-160	素域-163	素域-128	素域-256	素域-128
运算性能代价	+100.6%	+53.8%	+50%	+62%	+0
(点乘时间@频率/ms@MHz)	(0.34@194)	(0.6@316)	(82.5@8)	(0.089@222)	(0.37@312)
面积代价(面积/kGates)	+0(98)	+0(189)	+0(--)	+0(194.7)	+22.8%(360.3)
功耗代价(功率/mW)	--(34.4)	--(34.3)	--(--)	+50%(73.5)	+18.8%(120.0)
最小泄露轨迹数MTD 防护能力提升倍数	$ > {10^5}$ $ > 333$	$ > 2 \times {10^4}$ $ > 250$	$ > 5 \times {10^3}$ >119	-- --	$ > {10^4}$ $ > 312$

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