The aim of this paper is to study the threshold behavior for the satisfiability property of a random k-XOR-CNF formula or equivalently for the consistency of a random Boolean linear system with k variables per equation. For k ≥ 3 we show the existence of a sharp threshold for the satisfiability of a random k-XOR-CNF formula, whereas there are smooth thresholds for k=1 and k=2.

In this paper we study random linear systems with $k > 3$ variables per equation over the finite field GF(2), or equivalently k-XOR-CNF formulas. In a previous paper Creignou and Daudé proved that there exists a phase transition exhibiting a sharp threshold, for the consistency (satisfiability) of such systems (formulas). The control parameter for this transition is the ratio of the number of equations to the number of variables, and the scale for which the transition occurs remains somewhat elusive. In this paper we establish, for any $k > 3$, non-trivial lower and upper estimates of the value of the control ratio for which the phase transition occurs. For $k=3$ we get 0.89 and 0.93, respectively. Moreover, we give experimental results for $k=3$ suggesting that the critical ratio is about 0.92. Our estimates are clearly close to the critical ratio.

The Gaussian algorithm for lattice reduction in dimension 2 is analysed under its standard version. It is found that, when applied to random inputs in a continuous model, the complexity is constant on average, its probability distribution decays geometrically, and the dynamics are characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Analysis in the discrete model and detailed numerical data are also presented.