Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Material Type: Paper; Class: Prob Solving Strat in Math; Subject: Mathematics; University: Notre Dame; Term: Unknown 2003;
Typology: Papers
1 / 151
This page cannot be seen from the preview
Don't miss anything!
G´erard Maze, B.S., M.S.
Abstract
In this dissertation, we consider an extension of the discrete logarithm problem to the case of a semigroup acting on a finite set: the Semigroup Action Problem (SAP). New protocols and one-way trapdoor functions based on the difficulty of such problems are proposed. Several instances are studied both from a conceptual and cryptographic point of view. We discuss the application of existing generic algorithms to the resolution of an arbitrary SAP. The Pohlig-Hellman reduction leads to the notion of c-simplicity in semirings. Generic square-root at- tacks lead to semigroups with a negligible portion of invertible el- ements. After having described the situation when linear algebra over fields can be used, an application of the theory of finite c- simple semirings produces an example of SAP where no such known reduction applies. An extension of the Elliptic Curve Discrete Logarithm Problem (ECDLP) is defined using the Frobenius homomorphism of elliptic curves over finite fields. Actions induced by the Chebyshev polyno- mials are studied in different algebraic structures such as Fq , Z/nZ and Matn(Fq ). Those are shown to be equivalent to known hard problems such as FACTORING and DLP in finite fields. Finally, non-associative operations lead to the study of the SAP in Paige loops, i.e., finite simple non-associative Moufang loops.
To my parents, and to Sandrine.
iv
vii
x
xii
SAP Semigroup Action Problem
DHAP Diffie-Hellman semigroup action problem
DHP Diffie-Hellman problem
DLP discrete logarithm problem
ECDLP elliptic curve discrete logarithm problem
Fq the field with q elements
gcd greatest common divisor
lcm least common multiple
Matn(R) the set of n × n matrices over R
O(f (n)) function g(n) such that |g(n)| < c|f (n)| for some
constant c > 0 and all sufficiently large n
o(f (n)) function g(n) such that limn→∞ |g(n)|/|f (n)| = 0
RSA Rivest-Shamir-Adleman encryption scheme
xiii
Acknowledgement
I would like to thank here the people without whom these lines would not have been written. First and foremost, I am truly grate- ful to my advisor and friend, Joachim Rosenthal who helped me, supported me and gave me his entire trust. I am grateful to the Department of Mathematics of the Univer- sity of Notre Dame who provided me an excellent research environ- ment as well as the opportunity to finish my dissertation abroad. I am also grateful to Professor Charles Stuart who gave me the chance to work at the Ecole Polytechnique F´ed´erale de Lausanne while I was in Switzerland. I would like to thank the members of my defense committee, Karen Chandler, Claudia Polini and Andrew Sommese for their time and suggestions. I owe many thanks to Chris Monico who gave me precious advice as well as fruitful discussions. My thanks go out to my dear friends and colleagues Hugo, Tom, Aline, Elisa, Feride, Gregory, Marc-O. and Lionel. I would like to thank my family, maman, papa and Christine, for all their support and encouragement. Finally, Sandrine, ma douce Sandrine, merci.
xv
