Abstract An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of ‘hard supersingular curves’ that is equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular $\ell $-isogeny graph, which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd’s of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces, and applying Kummer surfaces and (v) using quantum random walks.

A linear error correcting code is a subspace of a finite-dimensional space over a finite field with a fixed coordinate system. Such a code is said to be locally recoverable with locality r if, for every coordinate, its value at a codeword can be deduced from the value of (certain) r other coordinates of the codeword. These codes have found many recent applications, e.g., to distributed cloud storage. We will discuss the problem of constructing good locally recoverable codes and present some constructions using algebraic surfaces that improve previous constructions and sometimes provide codes that are optimal in a precise sense. The main conceptual contribution of this paper is to consider surfaces fibered over a curve in such a way that each recovery set is constructed from points in a single fiber. This allows us to use the geometry of the fiber to guarantee the local recoverability and use the global geometry of the surface to get a hold on the standard parameters of our codes. We look in detail at situations where the fibers are rational or elliptic curves and provide many examples applying our methods.

We give a lower bound for the minimum distance of double circulant binary quadratic residue codes for primes p/spl equiv//spl plusmn/3(mod8). This bound improves on the square root bound obtained by Calderbank and Beenker, using a completely different technique. The key to our estimates is to apply a result by Helleseth, to which we give a new and shorter proof. Combining this result with the Weil bound leads to the improvement of the Calderbank and Beenker bound. For large primes p, their bound is of order /spl radic/(2p) while our new improved bound is of order 2/spl radic/p. The results can be extended to any prime power q and the modifications of the proofs are briefly indicated.

We give bounds for the minimal distance of duals of binary Bose-Chaudhuri-Hocquenghem (BCH) codes in a range where the Carlitz-Uchiyama bound is trivial. This is done by estimating the number of points on certain curves over finite fields.

We construct certain error-correcting codes over finite rings and estimate their parameters. These codes are constructed using plane curves and the estimates for their parameters rely on constructing "lifts" of these curves and then estimating the size of certain exponential sums.