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Public Key Cryptosystems from the Multiplicative Learning with Errors

We construct two public key cryptosystems based on the multiplicative learning with errors (MLWE) problem. The cryptosystem 1 (resp. 2) are semantically secure assuming the worst-case hardness of the decisional composite residuosity problem and the search (resp. decisional) LWE problem, which implies approximating the shortest vector problem in a lattice to small polynomial factors. Namely, our cryptosystems are secure as long as one of them is hard. Moreover, the cryptosystem 1 can be adapted to the chosen ciphertext secure cryptosystem. In addition, the cryptosystem 2 has the additively homomorphic property.