Elliptic and Hyperelliptic Curve Point Counting through Deformation

Point Counting on Elliptic and Hyperelliptic Curves

Point Counting on Genus 3 Non Hyperelliptic Curves

We propose an algorithm to compute the Frobenius polynomial of an ordinary non hyperelliptic curve of genus 3 over F2N . The method is a generalization of Mestre’s AGM-algorithm for hyperelliptic curves and leads to a quasi quadratic time algorithm for point counting. The current methods for point counting on curves over finite fields of small characteristic rely essentially on a p-adic approac...

Invalid-curve attacks on (hyper)elliptic curve cryptosystems

We extend the notion of an invalid-curve attack from elliptic curves to genus 2 hyperelliptic curves. We also show that invalid singular (hyper)elliptic curves can be used in mounting invalid-curve attacks on (hyper)elliptic curve cryptosystems, and make quantitative estimates of the practicality of these attacks. We thereby show that proper key validation is necessary even in cryptosystems bas...

The point counting problem for curves over finite fields

The group law on elliptic curves is well-known and gives rise to elliptic curve cryptography systems which find application to government and industry today. However, the generalisation to higher genus requires the manipulation of divisor classes rather than points, and analogues of key genus 1 results have yet to be found. Nonetheless, effective computation within the group is possible, and te...

Memory efficient hyperelliptic curve point counting

Let E be a hyperelliptic curve of genus g over a finite field of degree n and small characteristic. Using deformation theory we present an algorithm that computes the zeta function of E in time essentially cubic in n and quadratic memory. This improves substantially upon Kedlaya’s result which has the same time asymptotic, but requires cubic memory size. AMS (MOS) Subject Classification Codes: ...