On the evaluation of modular polynomials

We present two algorithms that, given a prime ell and an elliptic curve E/Fq, directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the j-invariants of the elliptic curves that are ell-isogenous to E. We do not assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be adapted to handle other types of modular polynomials, and we consider applications to point counting and the computation of endomorphism rings. We demonstrate the practical efficiency of the algorithms by setting a new point-counting record, modulo a prime q with more than 5,000 decimal digits, and by evaluating a modular polynomial of level ell = 100,019.Comment: 19 pages, corrected a typo in equation (8) and added equation (9

Correction to "On normal and subnormal q-ary codes"

No abstrac

Fast Computation of Special Resultants

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series

Cyclotomic Identity Testing and Applications

We consider the cyclotomic identity testing problem: given a polynomial $f(x_1,\ldots,x_k)$, decide whether $f(\zeta_n^{e_1},\ldots,\zeta_n^{e_k})$ is zero, for $\zeta_n = e^{2\pi i/n}$ a primitive complex $n$-th root of unity and integers $e_1,\ldots,e_k$. We assume that $n$ and $e_1,\ldots,e_k$ are represented in binary and consider several versions of the problem, according to the representation of $f$. For the case that $f$ is given by an algebraic circuit we give a randomized polynomial-time algorithm with two-sided errors, showing that the problem lies in BPP. In case $f$ is given by a circuit of polynomially bounded syntactic degree, we give a randomized algorithm with two-sided errors that runs in poly-logarithmic parallel time, showing that the problem lies in BPNC. In case $f$ is given by a depth-2 $\Sigma\Pi$ circuit (or, equivalently, as a list of monomials), we show that the cyclotomic identity testing problem lies in NC. Under the generalised Riemann hypothesis, we are able to extend this approach to obtain a polynomial-time algorithm also for a very simple subclass of depth-3 $\Sigma\Pi\Sigma$ circuits. We complement this last result by showing that for a more general class of depth-3 $\Sigma\Pi\Sigma$ circuits, a polynomial-time algorithm for the cyclotomic identity testing problem would yield a sub-exponential-time algorithm for polynomial identity testing. Finally, we use cyclotomic identity testing to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in coRNC: randomized NC with one-sided errors

Accelerating the CM method

Given a prime q and a negative discriminant D, the CM method constructs an elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X) modulo q. We consider an approach based on a decomposition of the ring class field defined by H_D, which we adapt to a CRT setting. This yields two algorithms, each of which obtains a root of H_D mod q without necessarily computing any of its coefficients. Heuristically, our approach uses asymptotically less time and space than the standard CM method for almost all D. Under the GRH, and reasonable assumptions about the size of log q relative to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D), which may be as small as O(|D|^(1/4)log q). The practical efficiency of the algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| > 10^15 and q ~ 2^33220. These examples are both an order of magnitude larger than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation and Mathematic

Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class

Integrable probability has emerged as an active area of research at the interface of probability/mathematical physics/statistical mechanics on the one hand, and representation theory/integrable systems on the other. Informally, integrable probabilistic systems have two properties: 1) It is possible to write down concise and exact formulas for expectations of a variety of interesting observables (or functions) of the system. 2) Asymptotics of the system and associated exact formulas provide access to exact descriptions of the properties and statistics of large universality classes and universal scaling limits for disordered systems. We focus here on examples of integrable probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality class and explain how their integrability stems from connections with symmetric function theory and quantum integrable systems.Comment: Proceedings of the ICM, 31 pages, 10 figure

Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles

We present a parallel algorithm for permanent mod 2^k of a matrix of univariate integer polynomials. It places the problem in ParityL subset of NC^2. This extends the techniques of [Valiant], [Braverman, Kulkarni, Roy] and [Bj\"orklund, Husfeldt], and yields a (randomized) parallel algorithm for shortest 2-disjoint paths improving upon the recent result from (randomized) polynomial time. We also recognize the disjoint paths problem as a special case of finding disjoint cycles, and present (randomized) parallel algorithms for finding a shortest cycle and shortest 2-disjoint cycles passing through any given fixed number of vertices or edges