基于类梅森数的物联网密钥交换快速取模算法

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

为适应物联网加密传输中大量轻型传感器节点运算性能和能源有限的特点,解决传感器运行RSA(Rivest-Shamir-Adleman)、DHM(Diffie-Hellman-Merkle)、Elgamal等公钥基础设施(PKI)加密算法所面临的运算速度、功耗等瓶颈问题,并简化相应的硬件加密电路逻辑设计,文中提出了一种基于类梅森数的密钥交换快速取模算法(CZ-Mod算法)。CZ-Mod算法利用梅森数的数学特性,使关键的mod(取模)运算的时间复杂度降至O(n)。首先,提出了一种以类梅森数为模数的快速取模运算mod1,使复杂的mod运算变成简单的二进制移位相加运算;其次,提出了一种以任意近似类梅森数的正整数为模数的快速取模运算mod2,在简化mod运算的同时扩大模数的取值范围;然后,对mod1、mod2运算作逻辑电路设计,以简化mod运算的硬件电路;最后,将以上工作应用到物联网节点的密钥交换中,以降低计算的复杂度,提高PKI加密算法的速度。实验测试结果表明:采用CZ-Mod算法的DHM密钥交换速度可达到常规算法的2.5~4倍;CZ-Mod算法精简,适合做物联网传感器的硬件电路设计。

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;CZMod algorithm is concise and fits the hardware circuit design for the IoT sensors.