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 -