Cryptanalysis of Rsa Using Algebraic and Lattice Methods

ii I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Studies: iii Preface We study the security of public key cryptosystems. In particular we study the RSA public key cryptosystem and several variants. We obtain our results using tools from the theory of integer lattices. We begin with an introduction to the theory of integer lattices and describe the main results to be used throughout the rest of the work. We then review the concept of a public key cryptosystem and in particular the RSA public key cryptosystem. Next we introduce a novel algorithm for the factorization of class of integers closely related to those used by RSA and other public key cryptosystems in the literature, showing that a new class of integers can be efficiently factored. We go on to introduce new attacks on the RSA public key cryptosystem which use partial knowledge of a user's secret key, showing that leaking one quarter of the bits of the secret key is sufficient to compromise RSA. Next we describe new attacks on the RSA public key cryptosystem when a short secret exponent is used. Lastly, we describe the three Sun-Yang-Laih key generation schemes for RSA, and introduce attacks to break two of these schemes. iv Acknowledgments It is impossible to thank enough everyone who has been of help over the past four years. It has been truly an honor and a pleasure to be a part of such a vibrant community of researchers, faculty, students, and friends, and I can make only a rough attempt to express my appreciation. I want to express my great gratitude to Dan Boneh, who has been both an advisor and a friend these many years. I would also like to thank my coauthors Yair Frankel, Nick Howgrave-Graham, and Phong Nguyen; working with them has been a great pleasure. During my time at Stanford I had the opportunity to interact with …