Fast Modulus Algorithm for Internet of Things Key Exchange Based on Mersenne-like Numbers

doi:10.12141/j.issn.1000-565X.220355

Abstract

Abstract:

In order to adapt to the limited computing performance and energy of numerous lightweight sensor nodes in the encrypted transmission of IoT (Internet of Things), this paper proposed a fast modulus algorithm (CZ-Mod algorithm) based on Mersenne-like numbers to slove the bottleneck problems of computing speed, power consumption and so on during the sensors run PKI (Public Key Infrastructure) encryption algorithms such as RSA (Rivest-Shamir-Adleman), DHM (Diffie-Hellman-Merkle), Elgamal, etc., and to simplify the corresponding hardware encrypting circuit logic design. CZ-Mod algorithm uses the mathematic characteristics of Mersenne numbers, and lowers the time complexity of its essential operation mod (modulo) into O(n). Firstly, a fast modulus algorithm mod1 using Mersenne-like numbers as modulus was presented, changing complex mod operation into simple binary shift/add operation; secondly, a fast modulus algorithm mod2 using any positive integers near Mersenne-like numbers as modulus was presented, expanding the modulus value range while simplifying mod operation; and then logic circuits of mod1 and mod2 operations were designed, simplifying mod operation hardware circuit. Finally, the above work was applied to the key exchange of IoT nodes, so as to lower the computing complexity and improve the speed of PKI encryption algorithms. The experiment test results indicate that the speed of DHM key exchange with CZ-Mod algorithm can reach 2.5 to 4 times of that of the conventional algorithm; CZ-Mod algorithm is concise and fits the hardware circuit design for the IoT sensors.

Key words: network security, sensor key, Mersenne number, encrypting operation circuit

CLC Number:

TP393

QIN Jiancheng, ZHONG Yu, CHENG Zhe, et al. Fast Modulus Algorithm for Internet of Things Key Exchange Based on Mersenne-like Numbers[J]. Journal of South China University of Technology(Natural Science Edition), 2023, 51(5): 24-35.

Figures/Tables 11

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算法	时间复杂度				数据预处理	占用空间/b
算法	取模	模乘	模幂	模幂（忽略分段乘法）	数据预处理	占用空间/b
传统mod	O（n²）	O（n²）	O（n³）	O（n³）	无	5n
蒙哥马利	O（n）	O（n²）	O（n³）	O（n²）	有	7n
CZ-Mod	O（n）	O（n²）	O（n³）	O（n²）	无	3n

微指令地址	微指令（控制信号）	下一微指令地址
1000	CLK_RegA， CLK_RegB， CLK_RegS	1001
1001	CLK_RegX0	1002
1002	！EN_ADD1，！EN_MUL1	C（n，n+1，…，n+m）=0？ 1003 ： 1001
1003	CLK_RegX	ffff

密钥位数	计时循环次数	单次DHM密钥交换时长/s			耗时倍数
密钥位数	计时循环次数	传统mod	mod1	mod2	耗时倍数
512	40 000	0.001 7		0.000 5	3.67
521	20 000	0.001 9	0.000 5		3.70
607	20 000	0.002 5	0.000 8		3.33
1 024	5 000	0.010 6		0.003 2	3.31
1 279	2 000	0.021 5	0.006 0		3.58
2 048	800	0.088 8		0.022 5	3.94
2 203	400	0.110 0	0.027 5		4.00
2 281	400	0.115 0	0.040 0		2.88
3 217	160	0.294 0	0.106 0		2.76
4 096	100	0.590 0		0.230 0	2.57
4 253	80	0.738 0	0.200 0		3.69
4 423	50	0.840 0	0.240 0		3.50
8 192	10	5.000 0		1.300 0	3.85
9 689	8	8.130 0	2.380 0		3.42
9 941	8	9.000 0	2.380 0		3.79

密钥位数	计时循环次数	单次DHM密钥交换时长/s			耗时倍数
密钥位数	计时循环次数	传统mod	mod1	mod2	耗时倍数
512	2 000	0.027 0		0.007 0	3.86
521	2 000	0.032 5	0.008 0		4.06
607	1 000	0.042 0	0.012 0		3.50
1 024	250	0.172 0		0.048 0	3.58
1 279	200	0.355 0	0.095 0		3.74
2 048	40	1.300 0		0.400 0	3.25
2 203	20	1.550 0	0.500 0		3.10
2 281	20	1.850 0	0.600 0		3.08
3 217	8	4.750 0	1.630 0		2.92
4 096	4	9.750 0		3.250 0	3.00
4 253	4	10.800 0	3.750 0		2.87
4 423	4	12.300 0	4.250 0		2.88
8 192	2	73.500 0		26.000 0	2.83
9 689	2	123.000 0	44.000 0		2.78
9 941	2	131.000 0	47.500 0		2.76

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