Cubic multivariate cryptosystems based on big field constructions and their vulnerability to a min-rank attack

In this work we analyze the security of cubic cryptographic constructions with respect to rank weakness. We detail how to extend the big field idea from quadratic to cubic, and show that the same rank defect occurs. We extend the min-rank problem and propose an algorithm to solve it in this setting. We show that for fixed small rank, the complexity is even lower than for the quadratic case. However, the rank of a cubic polynomial in n variables can be larger than n, and in this case the algorithm is very inefficient. We show that the rank of the differential is not necessarily smaller, rendering this line of attack useless if the rank is large enough. Similarly, the algebraic attack is exponential in the rank, thus useless for high rank.Resumen: En este trabajo analizamos la seguridad de construcciones criptogr´aficas c´ubicas con respecto a la debilidad del rango. Detallamos c´omo extender la idea de campo grande de cuadr´atico a c´ubico, y mostramos que la misma ca´ıda de rango ocurre. Extendemos el problema de rango m´ınimo y proponemos un algoritmo para resolverlo en este contexto. Mostramos que para rango bajo fijo, la complejidad es incluso m´as baja que en el caso cuadr´atico. Sin embargo, el rando de un polinomio c´ubico en n variables puede ser m´as grande que n, y en este caso el algoritmo es muy ineficiente. Mostramos que el rango del diferencial no es necesariamente m´as peque˜no, lo cual vuelve in´util esta l´ınea de ataque si el rango es lo suficientemente grande. Similarmente, el ataque algebr´aico es exponencial en el rango, y por lo tanto es in´util para rango alto.Maestrí

Algebraic geometry for tensor networks, matrix multiplication, and flag matroids

This thesis is divided into two parts, each part exploring a different topic within the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the so-called quantum Wielandt inequality, solving an open problem regarding the higher-dimensional version of matrix product states. Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which could potentially be used to prove new upper bounds on the complexity of matrix multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional algebras. Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian discriminant in terms of fundamental invariants. This answers Question 15 of the 27 questions on the cubic surface posed by Bernd Sturmfels. In the second part of this thesis, we apply algebro-geometric methods to study matroids and flag matroids. We review a geometric interpretation of the Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By generalizing Grassmannians to partial flag varieties, we obtain a new invariant of flag matroids: the flag-geometric Tutte polynomial. We study this invariant in detail, and prove several interesting combinatorial properties