6 research outputs found
Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module
Motivated by finding analogues of elliptic curve point counting techniques,
we introduce one deterministic and two new Monte Carlo randomized algorithms to
compute the characteristic polynomial of a finite rank-two Drinfeld module. We
compare their asymptotic complexity to that of previous algorithms given by
Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In
particular, we find that all three approaches represent either an improvement
in complexity or an expansion of the parameter space over which the algorithm
may be applied. Some experimental results are also presented
Drinfeld modules may not be for isogeny based cryptography
Elliptic curves play a prominent role in cryptography. For instance, the hardness of the elliptic curve discrete logarithm problem is a foundational assumption in public key cryptography. Drinfeld modules are positive characteristic function field analogues of elliptic curves. It is natural to ponder the existence/security of Drinfeld module analogues of elliptic curve cryptosystems. But the Drinfeld module discrete logarithm problem is easy even on a classical computer. Beyond discrete logarithms, elliptic curve isogeny based cryptosystems have have emerged as candidates for post-quantum cryptography, including supersingular isogeny Diffie-Hellman (SIDH) and commutative supersingular isogeny Diffie-Hellman (CSIDH) protocols. We formulate Drinfeld module analogues of these elliptic curve isogeny based cryptosystems and devise classical polynomial time algorithms to break these Drinfeld analogues catastrophically
Fast Algorithms for Finding the Characteristic Polynomial of a Rank-2 Drinfeld Module
This thesis introduces a new Monte Carlo randomized algorithm for computing the characteristic polynomial of a rank-2 Drinfeld module. We also introduce a deterministic algorithm that uses some ideas seen in Schoof's algorithm for counting points on elliptic curves over finite fields. Both approaches are a significant improvement over the current literature