TY - JOUR

T1 - A depth specific description of somewhat homomorphic encryption and its applications

AU - Lee, Hyang Sook

AU - Lim, Seongan

N1 - Publisher Copyright:
© 2015 NSP Natural Sciences Publishing Cor.

PY - 2015

Y1 - 2015

N2 - In this paper, we consider the depth-specific description of somewhat homomorphic encryption(SHE) schemes over integers. The ciphertexts of SHE scheme may have various forms depending on its encryption depth, and this makes the correctness check of the encryption scheme cumbersome. However, if one can present a SHE scheme depth-specifically, the correctness check is enough with depth-wise checks. We relate the homomorphic evaluation algorithms and binary operations on the set L of ciphertexts, and investigate what makes the depth-specific description is enough for a somewhat homomorphic encryption. We conclude that it is sufficient to have L with a ring-like structure with respect to the evaluation algorithms for a somewhat homomorphic encryption with relatively small depth. In fact, it is common to have the set of ciphertexts in a fully homomorphic encryption(FHE) scheme as a ring with respect to the evaluation algorithms. It is previously known that one can expand the message size of a SHE as t times larger with the ciphertexts t times larger using the Chinese Remainder Theorem(CRT). In this paper, we rewrite the message expansion method with CRT by using the depth specific description. Moreover, in the case of BGN cryptosystem, we show that one can expand the message size with smaller ciphertexts by using CRT twice. The rate of reduction of the ciphertext size depends on the security level. For example, for BGN cryptosystem using a bilinear group of 2048 bit, one can expand the size of plaintexts as t times larger with t/3 times larger ciphertexts. We see that the reducing rate becomes better if the security level increases.

AB - In this paper, we consider the depth-specific description of somewhat homomorphic encryption(SHE) schemes over integers. The ciphertexts of SHE scheme may have various forms depending on its encryption depth, and this makes the correctness check of the encryption scheme cumbersome. However, if one can present a SHE scheme depth-specifically, the correctness check is enough with depth-wise checks. We relate the homomorphic evaluation algorithms and binary operations on the set L of ciphertexts, and investigate what makes the depth-specific description is enough for a somewhat homomorphic encryption. We conclude that it is sufficient to have L with a ring-like structure with respect to the evaluation algorithms for a somewhat homomorphic encryption with relatively small depth. In fact, it is common to have the set of ciphertexts in a fully homomorphic encryption(FHE) scheme as a ring with respect to the evaluation algorithms. It is previously known that one can expand the message size of a SHE as t times larger with the ciphertexts t times larger using the Chinese Remainder Theorem(CRT). In this paper, we rewrite the message expansion method with CRT by using the depth specific description. Moreover, in the case of BGN cryptosystem, we show that one can expand the message size with smaller ciphertexts by using CRT twice. The rate of reduction of the ciphertext size depends on the security level. For example, for BGN cryptosystem using a bilinear group of 2048 bit, one can expand the size of plaintexts as t times larger with t/3 times larger ciphertexts. We see that the reducing rate becomes better if the security level increases.

KW - BGN cryptosytem

KW - Binary operation

KW - Chinese remainder theorem

KW - Homomorphic encryption

KW - Somewhat homomorphic encryption scheme

UR - http://www.scopus.com/inward/record.url?scp=84931363586&partnerID=8YFLogxK

U2 - 10.12785/amis/090329

DO - 10.12785/amis/090329

M3 - Article

AN - SCOPUS:84931363586

SN - 1935-0090

VL - 9

SP - 1345

EP - 1353

JO - Applied Mathematics and Information Sciences

JF - Applied Mathematics and Information Sciences

IS - 3

ER -