SuperSolver is an algorithm written in Sage/Python that solves the general supersingular isogeny problem using the Delfs-Galbraith algorithm. It accompanies the paper “SuperSolver: accelerating the Delfs-Galbraith algorithm with fast subfield root detection”, by Maria Corte-Real Santos, Craig Costello and Jia Shi.

When all of the files are in the Sage directory, the command

load("Runner.sage")

will perform the Delfs-Galbraith algorithm to find the isogeny between E1/GF(p^2) and E2/GF(p^2). There are default parameters in Runner.sage that can be modified to any instance of the supersingular isogeny problem. The code will generate new folders in the local directory for the reduced modular polynomials and solution.

The 6 inputs specified by the user are

p: the characteristic of the field GF(p^2)

j10: an element of GF(p)

j11: an element of GF(p)

j20: an element of GF(p)

j21: an element of GF(p)

supersolver: a boolean

The j-invariant of E1/GF(p^2) is j1=j10+j11.alpha and the j-invariant of E2/GF(p^2) is j2=j20+j21.alpha, where GF(p^2)=Fp(alpha) is the quadratic extension field specified in Runner.sage. Tiny primes (i.e. p<100) should be avoided since there might not be any supersingular j-invariants in GF(p^2)\GF(p), and the code assumes both input j-invariants are in GF(p^2); note that the supersingular isogeny problem is trivial for such small primes.

The supersolver boolean variable is true by default. Disabling it will call the original Delfs-Galbraith walk in the 2-isogeny graph, without the SuperSolver optimisations described in the paper above.

The file ExperimentAlgo0.sage can be run to obtain data on many instances of the problem for a given prime. After specifying a prime in ExperimentAlgo0.sage, the command

load("ExperimentAlgo0.sage")

followed by the command

Experiment(p,N)

will run N (an integer) instances of the subfield search problem for pseudo-random supersingular j-invariants over GF(p^2), as in the paper.

Please contact Maria Corte-Real Santos ([email protected]) and/or Craig Costello ([email protected]) and/or Jia Shi ([email protected]).

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