Research On FHE Scheme Over Matrix Rings

Fully homomorphic encryption was proposed by Rivest et al in early 1978.It refers to arbitrary calculation without the private key, and the calculation result is equivalent to the corresponding operation of the plaintext. It requires that one can compute addition and multiplication over the ciphertext and this two kinds of operations are homomorphic.In dddition, after any times of addition or multiplication operations, the ciphertext can still be correctly decrypted. The problem was solved by Gentry until 2009, and after that some fully homomorphic encryption schemes appeared.These fully homomorphic encryption schemes have a common feature which is constructed based on the existing mathematical hard computation problem, but most of the structure of these hard computational problems is commutative. It means that they cannot resist the quantum attacking. At present, fully homomorphic encryption schemes based on the non-commutative algebraic structure are too few, and these schemes contain defects on structure and construction. Therefore they are not widely used in real life. To design a new asymmetric fully homomorphic encryption scheme based on non-commutative algebra which can resist quantum attacking is very important. In view of this, we designed a fully homomorphic encryption scheme which can resist quantum attacking based on the non-commutative algebraic structure--matrix ring angle which based on the approximate matrix greatest common divisor (AMGCD).Fully homomorphic encryption can be used if there exists the untrusted third party, and the operation which untrusted third party operates on encrypted data do not affect the security of plaintext. This kind of special operation makes it has wide applications, such as cloud computing, biometrics, secure multi-party computation and electronic voting. But with the development of science and technology, it requires more focus on the capacity of cryptography scheme, and the requirment about the safety ciphertext data is more and more stringent. When ordinary cryptography scheme can’t resist quantum attacking, cryptography schemes based on non symmetric algebra structure will play its unique advantages to replace the ordinary cryptography scheme.In this thesis, we combined the AMGCD problems and decidable matrix equations problem. Firstly, we construct a somewhat homomorphism encryption scheme. With the operation increasing, the noise of the ciphertext is growing. We used the modual processing in noice processing,then the times of operation has no limit. Secondly, we analysed the scheme on the correctness,homomorphism and the number of homomorphic operation that this scheme can support. Finally, this scheme was used to solve the linear equations protocol of secure multi-party computation.