A New Algorithm Of Homomorphic Trapdoor Functions From SIS And Its Application

Motivated by the prevalence of cloud computing and big datas,Data Storage and Analysis of security are closely related to cryptography,The fully homomorphic encryption?FHE?can perform homomorphic computation over encrypted data without the knowledge of secret key,homomorphism cloud computing solutions meets the demand of the variety kinds of calculation in the cloud computing environment which supports addition and multiplication calculation,homomorphic encryption make it have high more efficiency and safety.Rencently.There are interested in the dual question of authenticity,Homomorphic signatures allow us to ascertain the authenticity of computation on out-sourced dat a.The development of electronic information technology requires more and more security to the cryptography,which leads to the establishment and de-velopment of new information technologies such as quantum.the research of Quantum information technology has accelerated the development of quan-tum mechanics and quantum system.Recently,anti-cryptography quantum computing research has become a hot topic.Lattice-based cryptography is an important field of cryptography,which security is based on NP-hard or NP-C problems,Therefore,Lattice-based cryptography is one of quantum resistant public key cryptographic systems.Gorbunov,Vaikuntanathan and Wichs[l]proposed homomorphic trapdoor function?HTDF?based on small integer solution?SIS?,In a Homomophic signa-ture scheme,given the public key and a vector of signatures ?:=??₁,?₂,…,?_l?over l messages x:?x₁,x₂,…x_l?based Homomorphic trapdoor function-s?HTDF?,there exists an efficient algorithem to produce a signature ?' for circuit g?x?.In Gorbunov-Vaikuntanathan-Wichs,scheme,which the maxi-mal depth d of the circuit is fixed a prior,the noise growth of the homomorphic operations depends on the size of the input xi and the boolean circuit g and the bound of U*is 2^??log??-d.they proved HTDF security by the knowledge of Entropy.The lattice-based homomorphic cryptography is not only valuable in cryp-tography retrieval,data storage and verification in cloud computing environ-ment,but also a quantum attack resistance.In this paper,the main research content:? Gadget-based trapdoor make it efficient for short bases as generic lattice trapdoors.We know it is easy to solve the SIS problem with respect to gadgets G.For a given vector v,There is a function G-1 which can find a short vector u so that Gu = v.We define a new algorithm of HTDF base on G-1 operation,which make the nosie of signatures is polynomial and greatly improve the depth of circuit.It is important for the design of fully homomorphic signatures scheme.? We prove the security of HTDF based on Gadget matrix?G?.Further-more,we analyze the security of homomorphic signatures from HTDF in Random Oracle model.We give the noise-level of signatures U*will be O?poly????d based on new algorithm.