1,040 research outputs found
Certification of Bounds of Non-linear Functions: the Templates Method
The aim of this work is to certify lower bounds for real-valued multivariate
functions, defined by semialgebraic or transcendental expressions. The
certificate must be, eventually, formally provable in a proof system such as
Coq. The application range for such a tool is widespread; for instance Hales'
proof of Kepler's conjecture yields thousands of inequalities. We introduce an
approximation algorithm, which combines ideas of the max-plus basis method (in
optimal control) and of the linear templates method developed by Manna et al.
(in static analysis). This algorithm consists in bounding some of the
constituents of the function by suprema of quadratic forms with a well chosen
curvature. This leads to semialgebraic optimization problems, solved by
sum-of-squares relaxations. Templates limit the blow up of these relaxations at
the price of coarsening the approximation. We illustrate the efficiency of our
framework with various examples from the literature and discuss the interfacing
with Coq.Comment: 16 pages, 3 figures, 2 table
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
We consider the problem of certifying an inequality of the form ,
, where is a multivariate transcendental function, and
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like , ,
, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the
European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250,
copyright EUCA 201
On formal verification of arithmetic-based cryptographic primitives
Cryptographic primitives are fundamental for information security: they are
used as basic components for cryptographic protocols or public-key
cryptosystems. In many cases, their security proofs consist in showing that
they are reducible to computationally hard problems. Those reductions can be
subtle and tedious, and thus not easily checkable. On top of the proof
assistant Coq, we had implemented in previous work a toolbox for writing and
checking game-based security proofs of cryptographic primitives. In this paper
we describe its extension with number-theoretic capabilities so that it is now
possible to write and check arithmetic-based cryptographic primitives in our
toolbox. We illustrate our work by machine checking the game-based proofs of
unpredictability of the pseudo-random bit generator of Blum, Blum and Shub, and
semantic security of the public-key cryptographic scheme of Goldwasser and
Micali.Comment: 13 page
Certification of Real Inequalities -- Templates and Sums of Squares
We consider the problem of certifying lower bounds for real-valued
multivariate transcendental functions. The functions we are dealing with are
nonlinear and involve semialgebraic operations as well as some transcendental
functions like , , , etc. Our general framework is to use
different approximation methods to relax the original problem into polynomial
optimization problems, which we solve by sparse sums of squares relaxations. In
particular, we combine the ideas of the maxplus estimators (originally
introduced in optimal control) and of the linear templates (originally
introduced in static analysis by abstract interpretation). The nonlinear
templates control the complexity of the semialgebraic relaxations at the price
of coarsening the maxplus approximations. In that way, we arrive at a new -
template based - certified global optimization method, which exploits both the
precision of sums of squares relaxations and the scalability of abstraction
methods. We analyze the performance of the method on problems from the global
optimization literature, as well as medium-size inequalities issued from the
Flyspeck project.Comment: 27 pages, 3 figures, 4 table
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