Common information techniques for the study of matroid representation and secret sharing schemes
- Autores: Michael Olugbenga Bamiloshin
- Directores de la Tesis: Oriol Farràs Ventura (dir. tes.)
- Lectura: En la Universitat Rovira i Virgili ( España ) en 2021
- Idioma: español
- Tribunal Calificador de la Tesis: Laszlo Csirmaz (presid.), Maria Bras Amorós (secret.), Andrei Romashchenko (voc.)
- Programa de doctorado: Programa de Doctorado en Ingeniería Informática y Matemáticas de la Seguridad por la Universidad Rovira i Virgili
- Materias:
- Enlaces
- Tesis en acceso abierto en: TDX
- Resumen
The main objective of this thesis was to provide new tools for the study of the twin problems of matroid representability and efficiency of secret sharing.
One of the most important problems in matroid theory is that of representation of matroids. Over time, different techniques have been applied to the study of this problem. They include, but are not limited to, using information and rank inequalities like the Ingleton inequality [Ing71]; checking the Ingleton-Main condition [InMa75]; and recently, using Frobenius flocks [Bollen18].
The common information (CI) property and the copy lemma are two tools that have been used to derive non-Shannon-type linear information and rank inequalities. Similar to the copy lemma is the Ahswelde-K\"orner (AK) information property. It is known that the CI property is satisfied by linearly representable matroids while all almost entropic matroids satisfy the AK property. In this thesis, we present two new methods for the study of matroid representation properties. For folded linear (i.e., multilinear) matroids, we develop the CI technique, while for almost entropic matroids we develop the AK technique.
We give a complete characterization of 8-point matroids with respect to folded linear representability and a near-complete characterization with respect to algebraicity and being almost entropic.
For matroids on 9-points and more, we needed complementary tools to the ones we have just described. For this, we turned to matroid intersection properties: the Euclidean intersection property, Levis intersection property, and the generalized Euclidean intersection property [BaKe88,BaWa89,AlHo95]. These are properties that are satisfied by linearly representable matroids. In this thesis, we develop recursive applications of these tools.
Using all the tools mentioned above, we found new non-representable matroids. We present a family of Ingleton-compliant matroids on 9 points having a configuration that prevents linear and folded-linear representability. This family includes the Tic-Tac-Toe matroid and hence, possibly adds to the family of potential matroids to solve the question of duality of algebraicity. We extend this family to include matroids on greater than 9 points. In addition, we find two other families of non folded-linear matroids.
The problem of representable matroids is one that carries over into the area of secret sharing in cryptography. Ideal access structures are ports of entropic matroids while ports of folded linear matroids admit ideal secret sharing schemes. Hence, determining which matroids satisfy which representation properties has implications for the study of efficiency measures in secret sharing.
Following along the work of Farr\`as et al. [FKMP20], we explore ports of matroids on 8 and as well 9 points. We give lower bounds on the information ratio for the ports of all matroids on 8 points both for linear schemes and for general schemes.
We also study ports of sparse-paving matroids in general. We found better general constructions and lower bounds on the information ratio for these matroid ports. Furthermore, we also show a separation result on lower bounds on the information ratio for non-Ingleton-compliant sparse-paving matroids.
References:
[AlHo95] Alfter, M. and Hochstättler, W.: On pseudomodular matroids and adjoints. Discrete Applied Mathematics, 60:3--11 (1995).
[BaKe88] Bachem, A. and Kern, W.: On sticky matroids. Discrete mathematics, 69(1):11--18 (1988).
[BaWa89] Bachem, A. and Wanka, A.: Euclidean intersection properties. Journal of Combinatorial Theory, Series B, 47(1):10--19 (1989).
[Bollen18] Bollen, G. P.: Frobenius flocks and algebraicity of matroids. Technische Universiteit Eindhoven (2018). PhD Thesis [FKMP18] Farràs, O., Kaced, T., Martín, S. and Padró, C.: Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing. Advances in Cryptology --- Eurocrypt 2018, Lecture Notes in Comput. Sci, 10820:597--621 (2018).
[Ing71] Ingleton, A. W.: Representation of matroids. In D. J. A Welsh, editor, Combinatorial Mathematics and its Applications, pages 149--167. Academic Press, London (1971).
[InMa75] Ingleton, A. W. and Main, R. A.: Non-algebraic matroids exist. Bull London Math Soc., 7:144--146 (1975).
