We prove that monotone circuits computing the perfect matching function on n-vertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.

The binary Euclidean algorithm is a variant of the classical Euclidean algorithm. It avoids multiplications and divisions, except by powers of two, so is potentially faster than the classical algorithm on a binary machine. We describe the binary algorithm and consider its average case behaviour. In particular, we correct some errors in the literature, discuss some recent results of Vallée, and describe a numerical computation which supports a conjecture of Vallée. 1

It is usually very hard, both for designers and users, to reason reliably about user interfaces. This article shows that ‘push button ’ and ‘point and click ’ user interfaces are algebraic structures. Users effectively do algebra when they interact, and therefore we can be precise about some important design issues and issues of usability. Matrix algebra, in particular, is useful for explicit calculation and for proof of various user interface properties. With matrix algebra, we are able to undertake with ease unusally thorough reviews of real user interfaces: this article examines a mobile phone, a handheld calculator and a digital multimeter as case studies, and draws general conclusions about the approach and its relevance to design.

The Bentley-Ottmann sweep-line method can be used to compute the arrangement of planar curves provided a number of geometric primitives operating on the curves are available. We discuss the mathematics of the primitives for planar algebraic curves of degree three or less and derive efficient realizations. As a result, we obtain a complete, exact, and efficient algorithm for computing arrangements of cubic curves. Conics and cubic splines are special cases of cubic curves. The algorithm is complete in that it handles all possible degeneracies including singularities. It is exact in that it provides the mathematically correct result. It is efficient in that it can handle hundreds of curves with a quarter million of segments in the final arrangement.

For any fixed dimension d, the linear programming problem with n inequality constraints can be solved on a probabilistic CRCW PRAM with O(n) processors almost surely in constant time. The algorithm always finds the correct solution. With nd/log² d processors, the probability that the algorithm will not finish within O(d² log² d) time tends to zero exponentially with n.

Abstract. In recent years there has been massive progress in the development of technologies for storing and processing of data. If statistical analysis could be applied to such data when it is distributed between several organisations, there could be huge benefits. Unfortunately, in many cases, for legal or commercial reasons, this is not possible. The idea of using the theory of multi-party computation to analyse efficient algorithms for privacy preserving data-mining was proposed by Pinkas and Lindell. The point is that algorithms developed in this way can be used to overcome the apparent impasse described above: the owners of data can, in effect, pool their data while ensuring that privacy is maintained. Motivated by this, we describe how to securely compute the mean of an attribute value in a database that is shared between two parties. We also demonstrate that existing solutions in the literature that could be used to do this leak information, therefore underlining the importance of applying rigorous theoretical analysis rather than settling for ad hoc techniques. 1

v Table of basic properties ix

We give an algorithm for modular composition of degree n univariate polynomials over a finite field Fq requiring n 1+o(1) log 1+o(1) q bit operations; this had earlier been achieved in characteristic n o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over Fq requiring (n 1.5+o(1) + n 1+o(1) log q) log 1+o(1) q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms. As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication. Supported by NSF DMS-0545904 (CAREER) and a Sloan Research Fellowship.

Abstract. In this work we present secure two-party protocols for various core problems in linear algebra. Our main result is a protocol to obliviously decide singularity of an encrypted matrix: Bob holds an n × n matrix, encrypted with Alice’s secret key, and wants to learn whether or not the matrix is singular (while leaking nothing further). We give an interactive protocol between Alice and Bob that solves the above problem in O(log n) communication rounds and with overall communication complexity of roughly O(n 2) (note that the input size is n 2). Our techniques exploit certain nice mathematical properties of linearly recurrent sequences and their relation to the minimal and characteristic polynomial of the input matrix, following [Wiedemann, 1986]. With our new techniques we are able to improve the round complexity of the communication efficient solution of [Nissim and Weinreb, 2006] from O(n 0.275) to O(log n). At the core of our results we use a protocol that securely computes the minimal polynomial of an encrypted matrix. Based on this protocol we exploit certain algebraic reductions to further extend our results to the problems of securely computing rank and determinant, and to solving systems of linear equations (again with low round and communication complexity). Keywords. Secure Linear Algebra, Linearly Recurrent Sequences, Wiedemann’s Algorithm. 1

The most efficient previously known parallel algorithms for the inversion of a nonsingular n x n matrix .4 or solving a linear system/Ix = b over the rational numbers require O(logn) time and M(n).x/ processors [provided that M(n) processors suffice in order to multiply two n x n rational matrices in time O(log n)]. Furthermore, the known polylog arithmetic time algorithms for those problems are numerically unstable. In this paper we apply Newton's iteration and initially choose an approximate inverse matrix by following Ben-Israel. This quadratically convergent and numerically stable iterative method takes O(logn) parallel time using M(n) processors to compute the inverse (within the relative prision 2 -" for a positive constant c) of an n x n rational matrix .4 with the condition number at most n d for a constant d. This is the optimum processor bound and by a factor of x// improvement of the previously known processor bounds for polylogarithmic time matrix inversion.'The algorithm does not require to precompute the condition number of the input matrix, but it just converges slower for ill-conditioned input matrices.