On random sampling of supersingular elliptic curves

We consider the problem of sampling random supersingular elliptic curves over finite fields of cryptographic size (SRS problem). The currently best-known method combines the reduction of a suitable complex multiplication (CM) elliptic curve and a random walk over some supersingular isogeny graph. Unfortunately, this method is not suitable when the endomorphism ring of the generated curve needs to be hidden, like in some cryptographic applications. This motivates a stricter version of the SRS problem, requiring that the sampling algorithm gives no information about the endomorphism ring of the output curve (cSRS problem). In this work we formally define the SRS and cSRS problems, which are both of theoretical interest. We discuss the relevance of the two problems for cryptographic applications, and we provide a self-contained survey of the known approaches to solve them. Those for the cSRS problem have exponential complexity in the characteristic of the base finite field (since they require computing and finding roots of polynomials of large degree), leaving the problem open. In the second part of the paper, we propose and analyse some alternative techniques—based either on the Hasse invariant or division polynomials—and we explain the reasons why they do not readily lead to efficient cSRS algorithms, but they may open promising research directions.