LWE-based Schemes Research Articles

Towards an efficient LWE‐based fully homomorphic encryption scheme

The security of most early fully homomorphic encryption schemes was based on the hardness of the Learning with Errors (LWE) problem. These schemes were inefficient in terms of per gate computations and public-key size. More efficient schemes were later developed based on the hardness of the Ring-LWE (RLWE) problem. While the hardness of the LWE problem is based on the hardness of the approximate shortest vector problem (GapSVPγ) over regular lattices, the hardness of the RLWE problem is based on the hardness of the approximate shortest vector problem over ideal lattices. As of now, it has not been proved that the (GapSVPγ) problem over ideal lattices is as difficult as the corresponding problem over regular lattices. In this work, the authors propose a multi-bit levelled fully homomorphic encryption scheme using multivariate polynomial evaluations whose security depends on the hardness of the LWE problem. In terms of per gate computation cost, this scheme is more efficient than existing LWE-based schemes. Further, for an appropriate choice of parameters, the per computation cost for homomorphic multiplication can be made asymptotically comparable to RLWE-based schemes in a parallel computing environment. For homomorphic multiplication, the scheme uses a polynomial-based technique that does not require relinearization (and key switching).

A public key encryption scheme based on a new variant of LWE with small cipher size

The lattice cryptosystem is considered to be able to resist the attacks of quantum computers. Lattice-based Public Key Encryption (PKE) schemes have attracted the interest of many researchers. In lattice-based cryptography, Learning With Errors (LWE) problem is a hard problem usually used to construct PKE scheme. To ensure the correctness of decryption, LWE-based schemes have a large ciphertext size. This makes these encryption schemes not practical enough when the communication bandwidth is limited. We propose a new variant of LWE, named Learning With Modulus (LWM) and prove that the new problem can be reduced from LWE problem. The proof idea of our reduction is similar to the reduction of LWR problem. We also construct a new PKE scheme based on the proposed LWM and LWE, which has small ciphertext size. For a 128 bits plaintext, the ciphertext size of our scheme is 53.57% of Lindner–Peikert’s (LP) scheme under the same security level. We use python to test the performance of our scheme. The results show that our scheme is only about 0.015 ms slower than LP in the decryption. The performance of our scheme for generating keys and encrypting messages is similar to LP.

사이버 물리 시스템의 보안을 위한 동형암호 소개

As attack strategies towards networks and digital controllers of cyber-physical systems (CPSs) develop, it has become difficult to ensure the security of physical plants such as factory facilities and power plants. Accordingly, encryption schemes have been adopted to enhance the security of CPSs, and among them, HE (Homomorphic Encryption) enables homomorphic operations between messages in encrypted state. Due to this nature of HE, there have been several approaches to encrypting controllers of CPSs. An encrypted controller refers to a controller which operates on entirely encrypted data including its states, parameters, and plant output, returning an encrypted control input. In this paper, we survey the properties and classification of HE and presents the advantages of encrypting controllers through HE. Moreover, we introduce various HE schemes such as Paillier cryptosystem, LWE-based scheme, and the cutting-edge scheme called HEAAN. Then, few examples of applying these schemes to control systems are introduced.

An automatic analysis approach toward indistinguishability of sampling on the LWE problem

Learning With Errors (LWE) is one of the Non-Polynomial (NP)-hard problems applied in cryptographic primitives against quantum attacks. However, the security and efficiency of schemes based on LWE are closely affected by the error sampling algorithms. The existing pseudo-random sampling methods potentially have security leaks that can fundamentally influence the security levels of previous cryptographic primitives. Given that these primitives are proved semantically secure, directly deducing the influences caused by leaks of sampling algorithms may be difficult. Thus, we attempt to use the attack model based on automatic learning system to identify and evaluate the practical security level of a cryptographic primitive that is semantically proved secure in indistinguishable security models. In this paper, we first analyzed the existing major sampling algorithms in terms of their security and efficiency. Then, concentrating on the Indistinguishability under Chosen-Plaintext Attack (IND-CPA) security model, we realized the new attack model based on the automatic learning system. The experimental data demonstrates that the sampling algorithms perform a key role in LWE-based schemes with significant disturbance of the attack advantages, which may potentially compromise security considerably. Moreover, our attack model is achievable with acceptable time and memory costs.

Learning-with-errors problem is easy with quantum samples

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE.

On the complexity of the BKW algorithm on LWE

This work presents a study of the complexity of the Blum–Kalai–Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension $$n \approx 250$$ when LWE is reduced to SIS. However, this assumes access to an unbounded number of LWE samples.