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Primitive Power Roots of Unity and Its Application to Encryption
Takato HIRANO Koichiro WADA Keisuke TANAKA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E92A
No.8
pp.18361844 Publication Date: 2009/08/01
Online ISSN: 17451337
DOI: 10.1587/transfun.E92.A.1836
Print ISSN: 09168508 Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications) Category: Theory Keyword: Paillier's encryption scheme, factoring, homomorphism, power roots of unity,
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Summary:
We first consider a variant of the SchmidtSamoaTakagi encryption scheme without losing additively homomorphic properties. We show that this variant is secure in the sense of INDCPA under the decisional composite residuosity assumption, and of OWCPA under the assumption on the hardness of factoring n=p^{2}q. Second, we introduce new algebraic properties "affine" and "preimage restriction," which are closely related to homomorphicity. Intuitively, "affine" is a tuple of functions which have a special homomorphic property, and "preimage restriction" is a function which can restrict the receiver to having information on the encrypted message. Then, we propose an encryption scheme with primitive power roots of unity in (Z/n^{s+1})^{}. We show that our scheme has, in addition to the additively homomorphic property, the above algebraic properties. In addition to the properties, we also show that the encryption scheme is secure in the sense of OWCPA and INDCPA under new number theoretic assumptions.

