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We define a pseudorandom function (PRF) F : K × X → Y to be bi-homomorphic when it is fully Key homomorphic and partially Input Homomorphic (KIH), i.e., given F (k1, x1) and F (k2, x2), there is an efficient algorithm to compute F (k1 ⊕ k2, x1 ⊖ x2), where ⊕ and ⊖ are (binary) group operations. The homomorphism on the input is restricted to a fixed subset of the input bits, i.e., ⊖ operates on some pre-decided m-out-of-n bits, where |x1| = |x2| = n, m < n, and the remaining n − m bits are identical in both inputs. In addition, the output length, ℓ, of the operator ⊖ is not fixed and is defined as n ≤ ℓ ≤ 2n, hence leading to Homomorphically induced Variable input Length (HVL) as n ≤ |x1 ⊖ x2| ≤ 2n. We present a learning with errors (LWE) based construction for a HVL-KIH-PRF family. Our construction is inspired by the key homomorphic PRF construction due to Banerjee and Peikert (Crypto 2014). We use our novel PRF family to construct an updatable encryption scheme, named QPC-UE-UU, which is quantum-safe, post-compromise secure and supports unidirectional ciphertext updates, i.e., the tokens can be used to perform ciphertext updates, but they cannot be used to undo completed updates. Our PRF family also leads to the first left/right key homomorphic constrained-PRF family with HVL.

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Anonymity is a desirable feature for vehicle-to-vehicle (V2V) communications, but it conflicts with other requirements such as non-repudiation and revocation. Existing, pseudonymbased V2V communications schemes rely on certificate generation and signature verification. These schemes require cumbersome key management, frequent updating of certificate chains and other costly procedures such as cryptographic pairings. In this paper, we present novel V2V communications schemes, that provide authentication, authorization, anonymity, non-repudiation, replay protection, pseudonym revocation, and forward secrecy without relying on traditional certificate generation and signature verification. Security and privacy of our schemes rely on hard problems in number theory. Furthermore, our schemes guarantee security and privacy in the presence of subsets of colluding malicious parties, provided that the cardinality of such sets is below a fixed threshold.

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