We apply Scholten’s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $$\mathbb {F}_{p^2}$$ and corresponding abelian surfaces over $$\mathbb {F}_{p}$$ . Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2, 2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography.

Lattice-based cryptography offers some of the most attractive primitives believed to be resistant to quantum computers. Following increasing interest from both companies and government agencies in building quantum computers, a number of works have proposed instantiations of practical post-quantum key exchange protocols based on hard problems in ideal lattices, mainly based on the Ring Learning With Errors (R-LWE) problem. While ideal lattices facilitate major efficiency and storage benefits over their non-ideal counterparts, the additional ring structure that enables these advantages also raises concerns about the assumed difficulty of the underlying problems. Thus, a question of significant interest to cryptographers, and especially to those currently placing bets on primitives that will withstand quantum adversaries, is how much of an advantage the additional ring structure actually gives in practice. Despite conventional wisdom that generic lattices might be too slow and unwieldy, we demonstrate that LWE-based key exchange is quite practical: our constant time implementation requires around 1.3ms computation time for each party; compared to the recent NewHope R-LWE scheme, communication sizes increase by a factor of 4.7x, but remain under 12 KiB in each direction. Our protocol is competitive when used for serving web pages over TLS; when partnered with ECDSA signatures, latencies increase by less than a factor of 1.6x, and (even under heavy load) server throughput only decreases by factors of 1.5x and 1.2x when serving typical 1 KiB and 100 KiB pages, respectively. To achieve these practical results, our protocol takes advantage of several innovations. These include techniques to optimize communication bandwidth, dynamic generation of public parameters (which also offers additional security against backdoors), carefully chosen error distributions, and tight security parameters.

Lattice-based cryptographic primitives are believed to offer resilience against attacks by quantum computers. We demonstrate the practicality of post-quantum key exchange by constructing cipher suites for the Transport Layer Security (TLS) protocol that provide key exchange based on the ring learning with errors (R-LWE) problem, we accompany these cipher suites with a rigorous proof of security. Our approach ties lattice-based key exchange together with traditional authentication using RSA or elliptic curve digital signatures: the post-quantum key exchange provides forward secrecy against future quantum attackers, while authentication can be provided using RSA keys that are issued by today's commercial certificate authorities, smoothing the path to adoption. Our cryptographically secure implementation, aimed at the 128-bit security level, reveals that the performance price when switching from non-quantum-safe key exchange is not too high. With our R-LWE cipher suites integrated into the Open SSL library and using the Apache web server on a 2-core desktop computer, we could serve 506 RLWE-ECDSA-AES128-GCM-SHA256 HTTPS connections per second for a 10 KiB payload. Compared to elliptic curve Diffie-Hellman, this means an 8 KiB increased handshake size and a reduction in throughput of only 21%. This demonstrates that provably secure post-quantum key-exchange can already be considered practical.

Cloud computing sparked interest in Verifiable Computation protocols, which allow a weak client to securely outsource computations to remote parties. Recent work has dramatically reduced the client's cost to verify the correctness of their results, but the overhead to produce proofs remains largely impractical. Geppetto introduces complementary techniques for reducing prover overhead and increasing prover flexibility. With Multi QAPs, Geppetto reduces the cost of sharing state between computations (e.g, For MapReduce) or within a single computation by up to two orders of magnitude. Via a careful choice of cryptographic primitives, Geppetto's instantiation of bounded proof bootstrapping improves on prior bootstrapped systems by up to five orders of magnitude, albeit at some cost in universality. Geppetto also efficiently verifies the correct execution of proprietary (i.e, Secret) algorithms. Finally, Geppetto's use of energy-saving circuits brings the prover's costs more in line with the program's actual (rather than worst-case) execution time. Geppetto is implemented in a full-fledged, scalable compiler and runtime that consume LLVM code generated from a variety of source C programs and cryptographic libraries.