Abstract We initiate the study of partial key exposure in Ring-LWE (RLWE)-based cryptosystems. Specifically, we (1) Introduce the search and decision Leaky R-LWE assumptions (Leaky R-SLWE, Leaky R-DLWE), to formalize the hardness of search/decision RLWE under leakage of some fraction of coordinates of the NTT transform of the RLWE secret. (2) Present and implement an efficient key exposure attack that, given certain 1/4-fraction of the coordinates of the NTT transform of the RLWE secret, along with samples from the RLWE distribution, recovers the full RLWE secret for standard parameter settings. (3) Present a search-to-decision reduction for Leaky R-LWE for certain types of key exposure. (4) Propose applications to the security analysis of RLWE-based cryptosystems under partial key exposure.
Abstract We initiate the study of partial key exposure in Ring-LWE (RLWE)-based cryptosystems. Specifically, we (1) Introduce the search and decision Leaky R-LWE assumptions (Leaky R-SLWE, Leaky R-DLWE), to formalize the hardness of search/decision RLWE under leakage of some fraction of coordinates of the NTT transform of the RLWE secret. (2) Present and implement an efficient key exposure attack that, given certain 1/4-fraction of the coordinates of the NTT transform of the RLWE secret, along with samples from the RLWE distribution, recovers the full RLWE secret for standard parameter settings. (3) Present a search-to-decision reduction for Leaky R-LWE for certain types of key exposure. (4) Propose applications to the security analysis of RLWE-based cryptosystems under partial key exposure.
Key Exchange is an essential primitive in cryptography that originated from the breakthrough idea of Diffie Hellman (DH) protocol presented in 1976.Following this work, there are many successful variants of DH using Elliptic curves, Isogeny and RLWE (Ring Learning with Errors).The Ring-Leaning with Errors (RLWE), Search and Decision version are hard problems in mathematics and it has been shown that solving worst case hard problems in ideal lattices can be reduced to solving RLWE in the average case.The reduction forms the basis of provable security for RLWE based key exchange against passive adversaries.The hardness of the decision version of RLWE, which means indistinguishability of RLWE samples from Uniform, makes it challenging to validate a RLWE form public key.This proves to be an advantage for malicious users to create public keys that deviate from a protocol to recover the secret of reused keys in Key Exchange (KE) protocols.In this thesis, we provide a cryptanalysis of RLWE key exchange, presenting two polynomial time strategies to exploit key reuse.Then we propose a defense against such exploits by presenting a Zero Knowledge authentication protocol to verify the prover's knowledge of a secret corresponding to his public key.The protocol is Statistical Zero Knowledge with negligible soundness and completeness errors so that the prover does not reveal any information about the secret in the proof.This is to ensure that a malicious verifier cannot impersonate the prover after the proof.The authentication protocol uses the signal function defined in RLWE key exchange to perform the verification.Firstly, I would like to express my gratitude to my advisor Prof.Ding for his expert advice, strong encouragement and support throughout my PhD study
This note describes a key exchange method based on the ring-LWE (RLWE)
assumption. It builds upon several results, including Regev's
landmark quantum reduction from certain worst case lattice problems
(approx. GapSVP and SIVP) to random instances of the search variant of
a particular learning problem (LWE). It also builds on the follow on
work of Lyubashevsky, Peikert and Regev on the average case hardness
of the RLWE search variant for polynomially bounded numbers of RLWE
samples, along with novel applications of automorphism groups in
number fields for a RLWE search to decision reduction (thereby
demonstrating pseudorandomness of RLWE in these number fields).
Subsequently, these results were adopted for the construction of
Diffie-Hellman like key exchange methods by Peikert, and then by
Lindner and Peikert followed by Ding and then by Ding, Xie and Lin who
proposed efficient variants of such protocols. Subsequent work by
Peikert proposed another efficient variant, phrased as a key
encapsulation method, incorporating a low bandwidth reconciliation
technique allowing two parties to exactly agree on a uniformly
distributed secret value from noisy RLWE instances. This was followed
by a concrete instantiation with parameter sets by Bos, Costello,
Naehrig, Stebila, followed by another instantiation by Alkim, Ducas,
Poppelmann and Schwabe with the same ring polynomial but a smaller
modulus and a different reconciliation method. Unlike most other
public key cryptography based key exchange methods, it is believed
that RLWE based key exchange would remain secure in the event that an
adversary is able to build a quantum computer. This document is a
product of the Crypto Forum Research Group (CFRG).
Ring Learning With Errors (RLWE)-based key exchange is one of the most efficient and secure primitive for post-quantum cryptography. One common approach to achieve key exchange over RLWE is error reconciliation. Recently, an efficient attack against reconciliation-based RLWE key exchange protocols with reused keys was proposed. This attack can recover a long-term private key if a key pair is reused. We also know that in the real world, key reuse is commonly adopted in applications like the Transport Layer Security (TLS) protocol to improve performance. Directly motivated by this attack, we construct a new randomized RLWE-based key exchange protocol against this attack. Our lightweight approach incorporates an additional ephemeral public error term into key exchange materials, so that this attack no longer works. With the same attack, we practically show that the signal value of our protocol is indistinguishable from uniform random, therefore, this attack no longer works. We explain how the attack fails, present 200-bit classic and 80-bit quantum secure parameter choice, efficient implementations, comparisons and discussion. Benchmark shows our protocol is truly efficient and even faster than related vulnerable protocols.
Current public key cryptosystems that are based on the hardness of integer factorization and discrete logarithm are insecure in the presence of large-scale quantum computers. Much effort has been devoted to replacing the quantum-insecure cryptosystems with newly developed "post-quantum" cryptosystem candidates, conjectured to be secure against quantum attack. Lattice-based cryptography has been widely recognized as a prominent candidate for practical post-quantum security.This dissertation improves the robustness and versatility of lattice-based cryptography through the following three contributions: 1. Chapter 3 introduces a constant-round protocol for unauthenticated group key exchange (i.e., with security against a passive eavesdropper). Group key exchange protocols allow a set of N parties to agree on a shared, secret key by communicating over a public network. Our protocol is based on the hardness of a lattice problem, which hence yields (plausible) post-quantum security. 2. In Chapter 4, we propose a framework for cryptanalysis of lattice-based schemes when certain types of information about the secret are leaked. Our framework generalizes the primal lattice reduction attack. The generalization allows for integrating the leaked information progressively before running a final lattice reduction step. Our framework can estimate the amount of security loss caused by the leaked information, and perform lattice reduction attacks with leaked information when computationally feasible. 3. Chapter 5 introduces an approach towards a ring analogue of the Leftover Hash Lemma (LHL). The LHL is a mathematical tool often used in the analysis of various lattice-based cryptosystems, as well as their leakage-resilient counterparts. However, it does not hold in the ring setting, which is typical for efficient cryptosystems. Lyubashevsky et al. (Eurocrypt '13) proved a "regularity lemma," which is used in the ring setting instead of the LHL; however, this applies only for centered, spherical Gaussian inputs, while the LHL applies when the input is drawn from any high min-entropy distribution. Our approach generalizes the "regularity lemma" of Lyubashevsky et al. to certain conditional distributions. A number of Ring-Learning with Errors based cryptosystems can achieve certain leakage resilience properties using our results.
In certain designs and applications of practical lattice-based cryptography, the use of a specialized variant of LWE problems, where the public matrix is sampled from a non-uniform distribution, is required to establish the security of the corresponding cryptographic schemes. Recently, the formal definition of LWE problems with semi-uniform seeds was introduced by the community, in which the hardness of Euclidean, ideal, and module lattice-based LWE problems with semi-uniform seeds was proved through reduction approach similar to those employed in the hardness proofs of entropic LWE problems. However, known reduction introduces significant losses in the Gaussian error parameters and lattice dimensions. Moreover, additional non-standard assumptions are required to demonstrate the hardness of LWE problems with semi-uniform seeds over rings. In this paper, a tighter reduction is proposed for LWE problems with semi-uniform seeds by incorporating modified techniques from the hardness proofs of Hint-LWE problems. The proposed reduction is unaffected by the algebraic structure of the underlying problems and can be uniformly applied to Euclidean, ideal, and module lattice-based LWE problems with semi-uniform seeds. The hardness of these LWE problems can be established based on standard LWE assumptions without the need for any additional non-standard assumption. Furthermore, the dimension of corresponding LWE problems remains unchanged, and the reduction introduces only minimal losses in Gaussian error parameters.