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We present a hash proof systems (HPS)based on hard lattice problems including learning with error (LWE) and inhomogeneous small integer solution (ISIS). Different from the HPS in past, ours is with error. That is to say, the deterministic private evaluation algorithm is not always equivalent to the deterministic public evaluation algorithm. The difference between them is one or zero. We also show how to construct a public-key encryption scheme which is semantically secure against key-leakagedoi:10.4156/ijact.vol4.issue8.1 fatcat:44gkrtfkvzcgngtjtbkk2jtmsm