Nemo/Hecke

Abstract

We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a lowlevel C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic. ACM Reference format:

On the construction of class fields

“…The algorithms outlined in the previous sections have been implemented in the number theory package Hecke [FHHJ17]. 1 As an application, we used our implementation to find number fields K having Galois closure L over ‫ޑ‬ with prescribed Galois group and such that K has minimal discriminant among all fields with this property.…”

Towards Practical Key Exchange from Ordinary Isogeny Graphs

“…This proof of concept also allowed us to estimate isogeny step costs, which we needed to generate the initial curve in §4. We developed several Julia packages 13 , built upon the computer algebra package Nemo [25]. Experiments were conducted using Julia 0.6 and Nemo 0.7.3 on Linux, with an Intel Core i7-5600U cpu at 2.60GHz.…”

Calcium

“…Most computer algebra systems arguably belong to one of two paradigms. Algebraic systems (Singular [9], Magma [3], Pari/GP [29], Sage [30], Nemo/Hecke [11], etc.) are designed for computation in definite algebraic structures, favoring strong data invariants.…”