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## On Constructing a Fully Homomorphic Encryption.

#### Abstract

Fully homomorphic encryption allows to perform calculations on private data, replacing them by computing the appropriate data in encrypted form. Gomomorphic encryption for one operation on integers is achieved, for example, by the RSA, El-Gamal, Goldwasser-Micali cryptosystems. Fully homomorphic encryption schemes that are gomomorphic over addition and multiplication were proposed recently. The first of them, using ideal lattices, was suggested by C. Gentry in 2009. Later C. Gentry and others offered another fully homomorphic encryption scheme with similar properties, but conducting operations on integers. The crucial part of these construction is the description of reencryption procedure. In this paper we proposed a new system for the re-encryption that looks like suggested by C. Gentry’s the scheme but, however, does not require the introduction of additional information about the secret key. In our scheme we use arbitrary representation inverse of private key that is represented using only logarithmic number on the length of input for possible distinct encryptions of bits. The proof uses the representation of the k-th bit of the integer sum of n bits as a symmetric polynomial on n variables of degree less then 2^k + 1.

#### Keywords

#### Edition

Proceedings of the Institute for System Programming, vol. 23, 2012, pp. 427-432.

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2012-23-25