We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of Õ(n1.5+ɛ)-approximation for lattice problems (such as GapSVP) under quantum reductions for any ɛ> 0 (or Õ(n2+ɛ)-approximation under classical reductions). This matches the best known hardness for “regular ” (non-homomorphic) lattice based public-key encryption up to the ɛ factor. A number of previous methods had hit a roadblock at quasipolynomial approximation. (As usual, a circular security assumption can be used to achieve a non-leveled FHE scheme.) Our approach consists of three main ideas: Noise-bounded sequential evaluation of high fan-in operations; Circuit sequentialization using Barrington’s Theorem; and finally, successive dimension-modulus reduction.
We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of Õ(n1.5+ɛ)-approximation for lattice problems (such as GapSVP) under quantum reductions for any ɛ> 0 (or Õ(n2+ɛ)-approximation under classical reductions). This matches the best known hardness for “regular ” (non-homomorphic) lattice based public-key encryption up to the ɛ factor. A number of previous methods had hit a roadblock at quasipolynomial approximation. (As usual, a circular security assumption can be used to achieve a non-leveled FHE scheme.) Our approach consists of three main ideas: Noise-bounded sequential evaluation of high fan-in operations; Circuit sequentialization using Barrington’s Theorem; and finally, successive dimension-modulus reduction.
We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of O(n1.5+ε)-approximation for lattice problems (such as GapSVP) under quantum reductions for any ε 〉 0 (or O(n2+ε)-approximation under classical reductions). This matches the best known hardness for "regular" (non-homomorphic) lattice based public-key encryption up to the ε factor. A number of previous methods had hit a roadblock at quasipolynomial approximation. (As usual, a circular security assumption can be used to achieve a non-leveled FHE scheme.)
In this paper, we first introduce a new concept of approximate lattice problem (ALP), which is an extension of learning with errors (LWE). Next, we propose two ALP-based public key encryption schemes. Then, we construct two new fully homomorphic encryption scheme (FHE) based on respectively approximate principal ideal lattice problem with related modulus (APIP-RM) and approximate lattice problem with related modulus (ALP-RM). Moreover, we also extend our ALP-RM-based FHE to the ALP problem with unrelated modulus (ALP-UM). Our work is different from previous works in three aspects: (1) We extend the LWE problem to the ALP problem. This ALP problem is similar to the closest vector problem in lattice. We believe that this problem is independent of interest. (2) We construct a new FHE by using a re-randomizing method, which is different from the squashing decryption in previous works. (3) The expansion rate is merely O(k) with k a security parameter in Our FHE, which can be improved to O(logk) by using dimension reduction [BV11], whereas all previous schemes are at least O(k*logk) [BV11, Gen11, LNV11]. Our method can also decrease a factor k of the expansion rate in their schemes.
Abstract: In this paper, we first introduce a new concept of approximate lattice problem (ALP), which is an extension of learning with errors (LWE). Next, we propose two ALP-based public key encryption schemes. Then, we construct two new fully homomorphic encryption scheme (FHE) based on respectively approximate principal ideal lattice problem with related modulus (APIP-RM) and approximate lattice problem with related modulus (ALP-RM). Moreover, we also extend our ALP-RM-based FHE to the ALP problem with unrelated modulus (ALP-UM). Our work is different from previous works in three aspects: (1) We extend the LWE problem to the ALP problem. This ALP problem is similar to the closest vector problem in lattice. We believe that this problem is independent of interest. (2) We construct a new FHE by using a re-randomizing method, which is different from the squashing decryption in previous works. (3) The expansion rate is merely O(k) with k a security parameter in Our FHE, which can be improved to O(logk) by using dimension reduction [BV11], whereas all previous schemes are at least O(k*logk) [BV11, Gen11, LNV11]. Our method can also decrease a factor k of the expansion rate in their schemes.
This paper proposes a generic construction of CCA-secure fully homomorphic encryption (FHE) scheme.First, we introduce a new primitive called multi-identity fully homomorphic encryption, which works in the multi-identity setting; that is, homomorphic evaluation can performed on ciphertexts created with different identities.Then, we propose an IND-sID-CPA secure multi-identity leveled FHE scheme over lattice.Finally, we present CCA-secure FHE from our proposed multi-identity FHE scheme and strongly EUF-CMA secure signature.The security analysis shows that our FHE scheme is CCA2-secure when the evaluation key is unavailable to the adversary, and remains CCA1-secure when the evaluation key is exposed in the standard model.
The idea of computing on encrypted data without decryption dates back to the notion of privacy homomorphisms introduced by Rivest, Adleman, and Dertouzos (1978). Their proposals built using the elegant structure of Chinese Remainder Theorem (CRT), were later shown to be insecure under simple known-plaintext attacks. Subsequent CRT-based fully homomorphic encryption (FHE) over the integers addresses this algebraic transparency by injecting noise and basing security on approximate common divisor–type assumptions, but the resulting designs are burdened by large public keys and costly ciphertext refresh procedures. In this work, we develop a new CRT-based FHE scheme whose security relies on the Ring-LWE (RLWE) hardness assumption. For this purpose, we introduce the CRT-RLWE problem. We show that the problem is at least as hard as the RLWE, thereby positioning our construction within the established post-quantum security landscape of RLWE-based cryptography. Our scheme retains an explicit CRT embedding, separating a message component modulo a prime-power plaintext modulus and an auxiliary CRT component, while using RLWE-style key and ring arithmetic for compactness and efficiency. Finally, we make it fully homomorphic by using a new bootstrapping procedure, that adopts the recryption paradigm for BGV/BFV schemes utilizing the linear transformation and digit extraction techniques.