Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

Chasing diagrams in cryptography

Cryptography is a theory of secret functions. Category theory is a general theory of functions. Cryptography has reached a stage where its structures often take several pages to define, and its formulas sometimes run from page to page. Category theory has some complicated definitions as well, but one of its specialties is taming the flood of structure. Cryptography seems to be in need of high level methods, whereas category theory always needs concrete applications. So why is there no categorical cryptography? One reason may be that the foundations of modern cryptography are built from probabilistic polynomial-time Turing machines, and category theory does not have a good handle on such things. On the other hand, such foundational problems might be the very reason why cryptographic constructions often resemble low level machine programming. I present some preliminary explorations towards categorical cryptography. It turns out that some of the main security concepts are easily characterized through the categorical technique of *diagram chasing*, which was first used Lambek's seminal `Lecture Notes on Rings and Modules'.Comment: 17 pages, 4 figures; to appear in: 'Categories in Logic, Language and Physics. Festschrift on the occasion of Jim Lambek's 90th birthday', Claudia Casadio, Bob Coecke, Michael Moortgat, and Philip Scott (editors); this version: fixed typos found by kind reader

The geometry of efficient arithmetic on elliptic curves

The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripling, by considering families of elliptic curves admiting a $2$-torsion or $3$-torsion point

Point compression for the trace zero subgroup over a small degree extension field

Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation for such points. Our representation is compatible with scalar multiplication. We give a point compression algorithm to compute the representation and a decompression algorithm to recover the original point (up to some small ambiguity). The algorithms are efficient for trace zero varieties coming from small degree extension fields. We give explicit equations and discuss in detail the practically relevant cases of cubic and quintic field extensions.Comment: 23 pages, to appear in Designs, Codes and Cryptograph

Cryptography and number theory in the classroom -- Contribution of cryptography to mathematics teaching

Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts

Recognizing and Reducing Ambiguity in Mathematics Curriculum and Relations of θ-Functions in Genus One and Two: A Geometric Perspective

Anxiety and mathematics come hand in hand for many individuals. This is due, inpart, to the fact that the only experience they have with mathematics is what somemathematics educators refer to as schoolmath, which uses a somewhat differentlanguage than real mathematics. The language of schoolmath can cause individu-als to have confusion and develop misconceptions related to several mathematicalconcepts. One such concept is a fraction. In chapter one of this report, one possiblereason for this is discussed and a possible solution is purposed.In chapter three of this report, genus-two curves admitting an elliptic involutionare related to pairs of genus-one curves. This classical work dates back to early 20thcentury and is known as Jacobi reduction. Jacobians of genus-two curves can beused to construct complex two-dimensional complex projective manifolds knownas Kummer surfaces. On the other hand, the defining coordinates and parameters ofelliptic curves and Kummer surfaces can be related to Jacobi θ-functions and Siegelθ-functions, respectively. This result goes back to the seminal work of Mumford inthe 1980s. We use a geometric relation between elliptic curves and Kummer surfacesto derive functional relations between θ-functions