Elliptic curves have become key ingredients for instantiating zero-knowledge proofs and more generally proof systems. Recently, there have been many tailored constructions of these curves that aim at efficiently implementing different kinds of proof systems. In this survey we provide the reader with a comprehensive overview on existing work and revisit the contributions in terms of efficiency and security. We present an overview at three stages of the process: curves to instantiate a SNARK, curves to instantiate a recursive SNARK, and also curves to express an elliptic-curve related statement. We provide new constructions of curves for SNARKs and generalize the state-of-the-art constructions for recursive SNARKs. We also exhaustively document the existing work and open-source implementations.

The generation of random numbers is a keystone function in any cryptographic protocol. Indeed, in a security context, the random numbers generation shall withstand assaults from adversaries. It is thus paramount to validate both its functionality and its robustness in front of attacks, including fault injection attacks. The verification implies tests, which shall thus be carried out in nominal but also in perturbed operational environments. In this paper, we review standard tests already existing and still under development. As a first contribution, we suggest a new kind of metrics to assess the quality of the random sequences of bits. As a second contribution, we analyse fault injections in true random number generators and explore whether such faulted behavior can be self-induced within the circuit itself. This analysis reveals a plausible interpretation of the behavior of circuits based on the analysis of long term noise, e.g., TRNGs based on ring oscillators.