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.

Most recent theoretical literature on program obfuscation is based on notions like virtual black box (VBB) obfuscation and indistinguishability obfuscation (iO). These notions are very strong and are hard to satisfy. Further, they offer far more protection than is typically required in practical applications. On the other hand, the security notions introduced by software security researchers are suitable for practical designs but are not formal or precise enough to enable researchers to provide a quantitative security assurance. Hence, in this paper, we introduce a new formalism for practical program obfuscation that still allows rigorous security proofs. We believe our formalism will make it easier to analyse the security of obfuscation schemes. To show the flexibility and power of our formalism, we give a number of examples. Moreover, we explain the close relationship between our formalism and the task of providing obfuscation challenges.