Many issues have algorithmic aspects. The corresponding results are integrated into the open source mathematics software SageMath. This also applies to essential tools, meaning that SageMath modules for both finite state machines and for asymptotic expansions (supported by Google Summer of Code) have been created.
Our field of activity ranges from a broad field in discrete mathematics, including algebra and number theory (permutation polynomials, elliptic curves, digit expansions), graph theory (extremal graphs relating to graph theoretical indices) to analysis of algorithms and analytic combinatorics (regular sequences, trees, lattice paths) for use in cryptography.
A prototypical example is the analysis of digit expansions and how they can be used for the efficient implementation of scalar multiplication in abelian groups, for example in elliptic curve cryptography. Digit expansions are not only based on rational integer bases, but also algebraic integer bases, which correspond to efficiently calculable endomorphisms of the curve (such as the Frobenius endomorphism).
Redundant digit sets are selected in order to achieve expansions with lower weight and therefore better running times. The relevant issues include selecting a digit set, existence and minimality of the expansion along with the asymptotic and probabilistic analysis of the weight obtained.
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