Pierre Valarcher

Extending the loop language with higher-order procedural variables

We extend Meyer and Ritchies Loop language with higher-order procedures and procedural variables and we show that the resulting programming language (called Loop^ω) is a natural imperative counterpart of Gödel System T. The argument is two-fold: (1) we define a translation of the Loop^ω language into System T and we prove that this translation actually provides a lock-step simulation, (2) using a converse translation, we show that Loop^ω is expressive enough to encode any term of System T. Moreover, we define the “iteration rank” of a Loop^ω program, which corresponds to the classical notion of “recursion rank” in System T, and we show that both translations preserve ranks. Two applications of these results in the area of implicit complexity are described.

HFE and BDDs: A Practical Attempt at Cryptanalysis

HFE (Hidden Field Equations) is a public key cryptosystem using univariate polynomials over finite fields. It was proposed by J. Patarin in 1996. Well chosen parameters during the construction produce a system of quadratic multivariate polynomials over \({\mathbb{F}_2}\) as the public key. An enclosed trapdoor is used to decrypt messages. We propose a ciphertext-only attack which mainly consists in satisfying a boolean formula. Our algorithm is based on BDDs (Binary Decision Diagrams), introduced by Bryant in 1986, which allow to represent and manipulate, possibly efficiently, boolean functions. This paper is devoted to some experimental results we obtained while trying to solve the Patarin’s challenge. This approach was not successful, nevertheless it provided some interesting information about the security of HFE cryptosystem.

Intensional Semantics of System T of Gödel

Abstract This paper is a contribution to the development of a theory of behaviour of programs. We study an intensional behaviour of system T of Godel that is devoted to capturing not which function is computed but how it is computed. This intensional behaviour is captured by a denotational semantics in the domain of lazy natural numbers. It is shown that all sequential algorithms definable in system T are intensional behaviours. This leads us to obtain a general representation theorem asserting that we may compute every definable function of system T with the behaviour of a sequential algorithm using a higher-order term.