对称三进制编码的同态加密算术运算研究

Homomorphically Encrypted Arithmetic Operations over Symmetric Ternary Coding

全同态加密可以在不泄露明文信息的情况下,支持对密文的任意函数运算并在解密后得到与在明文上执行相应运算一致的结果.全同态加密应用需要先通过编码将待处理的数据编码为明文串,然后将编码得到的明文同态加密为密文并进行密文上的函数运算,再将运算得到的密文进行同态解密,最后将解密结果解码为应用中的数据类型.不同的编码方案对全同态加密应用的效率有较大的影响.本文将对称三进制编码方法应用于全同态加密应用,设计了整数的对称三进制编码算术运算方法.同时,分析了对称三进制编码下的整数加法、整数减法和整数乘法的位乘法深度.设计的整数对称三进制编码算术运算不需要对编码设置符号位,并且按照减少位乘法运算次数和深度的原则对运算过程进行了优化,最后,利用HElib库实现了整数的对称三进制编码下10个比特位的同态加法、同态减法和同态乘法的算术运算,计算期间不需要Bootstrapping过程.实验结果表明,整数在对称三进制编码下的同态算术运算效率优于二进制编码.

Fully homomorphic encryption can support the operation of any function of the ciphertext obtaining the result of the corresponding operation on the plaintext after decryption. Fully homomorphic encryption application needs to encode the data into plaintext strings by encoding, and then homomorphic encrypts the plaintext into a ciphertext and performs the function operation on the ciphertext. Then, decrypts the ciphertext, and finally decode the decryption results into the data type in the application. Different coding schemes have a great influence on the efficiency of fully homomorphic encryption applications. In this paper, the symmetric ternary coding method is applied to the fully homomorphic encryption application, and an integer symmetric ternary coding algorithm is designed. Meanwhile, the bitwise multiplication depth of integer addition, integer subtraction and integer multiplication under symmetric ternary coding is analyzed. Designed integer symmetric ternary coding arithmetic operations do not require the setting of the sign bits for the code, and the operation process is optimized according to the principle of reducing the number of bitwise multiplication depth.Finally, we use the HElib library to achieve the integer symmetric ternary encoding under the 10-bit homomorphism addition, homomorphic subtraction and homomorphism multiplication arithmetic, and the calculation period does not need Bootstrapping process. The experimental results show that the homomorphic arithmetic efficiency of integers under symmetric ternary coding is superior to binary coding.