The Research And Design Of Delegation Of Computation Based On Matrix Homomorphic Encryption Scheme

Delegation of computation of Linear Algebra has great significance on theoreticalresearch and applications. With computer technology towards the development of amultipolar direction, computing resources distribute increasingly uneven, but delegation ofcomputation technology can make computing resources distribution more reasonable andmore flexible, so it is important for us to study on delegation of computation. The researchon delegation of computation is still in the phase of theory currently. The biggest defect ofdelegation of computation scheme based on fully homomorphic encryption is its deficiency,and impractical of fully homomorphic encryption is becoming an obstacle of practice ofdelegation of computation. In allusion to this problem, we have the following researchresults:i. In order to make matrix posses homomorphic characters, we first definehomomorphic encryption over matrix. We construct a homomorphic encryption schemeover matrix using the hardness of approximate greatest common divisor after doing a lot ofresearches on the computation over matrices. This scheme satisfies CPA security, itsatisfies additive homomorphic operation many times and at least one time multiplicationhomomorphic operation. It’s a practical scheme.ii. Huge ciphertext and deficiency of computation are two shortcomings ofHomomorphic encryption over matrices based on homomorphic encryption scheme. Tospeed up the operation on cipher matrices, and make matrix operation fully homomorphic,we present an efficient fully homomorphic encryption over matrix scheme. Based on theinstinct deficiency of homomorphic scheme, we encrypt the whole matrix instead ofencrypting every element of matrix using homomorphic encryption scheme traditionally.We construct a secure fully homomorphic encryption over matrix scheme, which has highrate of efficiency. iii. We implement the above two scheme based on open resources, and do some tests onthe encryption system to get some truly data and verify the feasibility and practice of thescheme.iv. We compare the above two schemes with GHV scheme at all respects based on thetests data, and choose fully homomorphic encryption over matrix scheme as the foundationto construct delegation of computation of linear algebra scheme finally. The solvedproblems include delegation of matrix compound computation, delegation of matrix setaddition, delegation of computing special matrix’s solution and delegation of computingmatrix roots. We analysis and demonstrate the security of those schemes, and give anapplication of the scheme.