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Key Exchange (KE) is, undoubtedly, one of the most used cryptographic primitives in practice. Its authenticated version, Authenticated Key Exchange (AKE), avoids man-in-the-middle-based attacks by providing authentication for both parties involved. It is widely used on the Internet, in protocols such as TLS or SSH. In this work, we provide new constructions for KE and AKE based on ideal lattices in the Random Oracle Model (ROM). The contributions of this work can be summarized as follows: 1) It is well-known that RLWE-based KE protocols are not robust for key reuses since the signal function leaks information about the secret key. We modify the design of previous RLWE-based KE schemes to allow key reuse in the ROM. Our construction makes use of a new technique called pasteurization which enforces a supposedly RLWE sample sent by the other party to be indeed indistinguishable from a uniform sample and, therefore, ensures no information leakage in the whole KE process. 2) We build a new AKE scheme based on the construction above. The scheme provides implicit authentication (that is, it does not require the use of any other authentication mechanism, like a signature scheme) and it is proven secure in the Bellare-Rogaway model with weak Perfect Forward Secrecy in the ROM. It improves previous designs for AKE schemes based on lattices in several aspects. Our construction just requires sampling from only one discrete Gaussian distribution and avoids rejection sampling and noise flooding techniques, unlike previous proposals (Zhang et al., EUROCRYPT 2015). Thus, the scheme is much more efficient than previous constructions in terms of computational and communication complexity. Since our constructions are provably secure assuming the hardness of the RLWE problem, they are considered to be robust against quantum adversaries and, thus, suitable for post-quantum applications.

We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between (1) The r-th central derivative of the quadratic base change L-function associated to an everywhere unramified cuspidal automorphic representation π of PGL2; (2) The self-intersection number of the π-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of L-functions.

Anonymous veto networks (AV-nets), originally proposed by Hao and Zielinski (2006), are particularly lightweight protocols for evaluating a veto function in a peer-to-peer network such that anonymity of all protocol participants is preserved. Prior to this work, anonymity in all AV-nets from the literature relied on the decisional Diffie-Hellman (DDH) assumption and can thus be broken by (scalable) quantum computers. In order to defend against this threat, we propose two practical and completely lattice-based AV-nets. The first one is secure against passive and the second one is secure against active adversaries. We prove that anonymity of our AV-nets reduces to the ring learning with errors (RLWE) assumption. As such, our AV-nets are the first ones with post-quantum anonymity. We also provide performance benchmarks to demonstrate their practicality.