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Complexity of Lattice Problems
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We propose multi-bit versions of several single-bit cryptosystems based on lattice problems, the error-free version of the Ajtai-Dwork cryptosystem by Goldreich, Goldwasser, and Halevi [CRYPTO '97], the Regev cryptosystems [JACM 2004 and STOC 2005], and the Ajtai cryptosystem [STOC 2005]. We develop a universal technique derived from a general structure behind them for constructing their multi-bit versions without increase in the size of ciphertexts. By evaluating the trade-off between the decryption errors and the hardness of underlying lattice problems, it is shown that our multi-bit versions encrypt O(log n)-bit plaintexts into ciphertexts of the same length as the original ones with reasonable sacrifices of the hardness of the underlying lattice problems. Our technique also reveals an algebraic property, named pseudohomomorphism, of the lattice-based cryptosystems.