International Association
for Cryptologic Research

$$\mathsf {LWE}$$ LWE -based key-exchange protocols lie at the heart of post-quantum public-key cryptography. However, all existing protocols either lack the non-interactive nature of Diffie–Hellman key exchange or polynomial $$\mathsf {LWE}$$ LWE -modulus, resulting in unwanted efficiency overhead. We study the possibility of designing non-interactive $$\mathsf {LWE}$$ LWE -based protocols with polynomial $$\mathsf {LWE}$$ LWE -modulus. To this end, we identify and formalize simple non-interactive and polynomial $$\mathsf {LWE}$$ LWE -modulus variants of the existing protocols, where Alice and Bob simultaneously exchange one or more (ring) $$\mathsf {LWE}$$ LWE samples with polynomial $$\mathsf {LWE}$$ LWE -modulus and then run individual key reconciliation functions to obtain the shared key. We point out central barriers and show that such non-interactive key-exchange protocols are impossible in either of the following cases: (1) the reconciliation functions first compute the inner product of the received $$\mathsf {LWE}$$ LWE sample with their private $$\mathsf {LWE}$$ LWE secret. This impossibility is information theoretic. (2) One of the reconciliation functions does not depend on the error of the transmitted $$\mathsf {LWE}$$ LWE sample. This impossibility assumes hardness of $$\mathsf {LWE}$$ LWE . We show that progress toward either a polynomial $$\mathsf {LWE}$$ LWE -modulus $$\text {NIKE}$$ NIKE construction or a general impossibility result has implications to the current understanding of lattice-based cryptographic constructions. Overall, our results show possibilities and challenges in designing simple (ring) $$\mathsf {LWE}$$ LWE -based non-interactive key-exchange protocols.

$$mathsf {LWE}$$ based key-exchange protocols lie at the heart of post-quantum public-key cryptography. However, all existing protocols either lack the non-interactive nature of Diffie-Hellman key-exchange or polynomial $$mathsf {LWE}$$ -modulus, resulting in unwanted efficiency overhead. We study the possibility of designing non-interactive $$mathsf {LWE}$$ -based protocols with polynomial $$mathsf {LWE}$$ -modulus. To this end, We identify and formalize simple non-interactive and polynomial $$mathsf {LWE}$$ -modulus variants of existing protocols, where Alice and Bob simultaneously exchange one or more (ring) $$mathsf {LWE}$$ samples with polynomial $$mathsf {LWE}$$ -modulus and then run individual key reconciliation functions to obtain the shared key. We point out central barriers and show that such non-interactive key-exchange protocols are impossible if: (1) the reconciliation functions first compute the inner product of the received $$mathsf {LWE}$$ sample with their private $$mathsf {LWE}$$ secret. This impossibility is information theoretic. (2) One of the reconciliation functions does not depend on the error of the transmitted $$mathsf {LWE}$$ sample. This impossibility assumes hardness of $$mathsf {LWE}$$ . We give further evidence that progress in either direction, of giving an $$mathsf {LWE}$$ -based $$mathrm {NIKE}$$ protocol or proving impossibility of one will lead to progress on some other well-studied questions in cryptography. Overall, our results show possibilities and challenges in designing simple (ring) $$mathsf {LWE}$$ -based non-interactive key exchange protocols.