International Association
for Cryptologic Research

We provide explicit descriptions for radical 2-isogenies in dimensions one, two and three using theta coordinates. These formulas allow us to efficiently navigate in the corresponding isogeny graphs. As an application of this, we implement different versions of the CGL hash function. Notably, the three-dimensional version is fastest, which demonstrates yet another potential of using higher dimensional isogeny graphs in cryptography.

Following Ibukiyama, Katsura and Oort, all principally polarized superspecial abelian surfaces over $\overline{\mathbb{F}}_p$ can be represented by a certain type of $2 \times 2$ matrix $g$, having entries in the quaternion algebra $B_{p,\infty}$. We present a heuristic polynomial-time algorithm which, upon input of two such matrices $g_1, g_2$, finds a "connecting matrix" representing a polarized isogeny of smooth degree between the corresponding surfaces. Our algorithm should be thought of as a two-dimensional analog of the KLPT algorithm from 2014 due to Kohel, Lauter, Petit and Tignol for finding a connecting ideal of smooth norm between two given maximal orders in $B_{p, \infty}$. The KLPT algorithm has proven to be a versatile tool in isogeny-based cryptography, and our analog has similar applications; we discuss two of them in detail. First, we show that it yields a polynomial-time solution to a two-dimensional analog of the so-called constructive Deuring correspondence: given a matrix $g$ representing a superspecial principally polarized abelian surface, realize the latter as the Jacobian of a genus-$2$ curve (or, exceptionally, as the product of two elliptic curves if it concerns a product polarization). Second, we show that, modulo a plausible assumption, Charles-Goren-Lauter style hash functions from superspecial principally polarized abelian surfaces require a trusted set-up. Concretely, if the matrix $g$ associated with the starting surface is known then collisions can be produced in polynomial time. We deem it plausible that all currently known methods for generating a starting surface indeed reveal the corresponding matrix. As an auxiliary tool, we present an efficient method for converting isogenies of powersmooth degree into the corresponding connecting matrix, a step for which a previous approach by Chu required super-polynomial (but sub-exponential) time.

It has recently been rigorously proven (and was previously known under certain heuristics) that the general supersingular isogeny problem reduces to the supersingular endomorphism ring computation problem. However, in order to attack SIDH-type schemes, one requires a particular isogeny which is usually not returned by the general reduction. At Asiacrypt 2016, Galbraith, Petit, Shani and Ti presented a polynomial-time reduction of the problem of finding the secret isogeny in SIDH to the problem of computing the endomorphism ring of a supersingular elliptic curve. Their method exploits the fact that secret isogenies in SIDH are of degree approximately $p^{1/2}$. The method does not extend to other SIDH-type schemes, where secret isogenies of larger degree are used and this condition is not fulfilled. We present a more general reduction algorithm that generalises to all SIDH-type schemes. The main idea of our algorithm is to exploit available torsion point images together with the KLPT algorithm to obtain a linear system of equations over a certain residue class ring. We show that this system will have a unique solution that can be lifted to the integers if some mild conditions on the parameters are satisfied. This lift then yields the secret isogeny. One consequence of this work is that the choice of the prime $p$ in \mbox{B-SIDH} is tight.