We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.

We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in$\mathbb{F}_{p^{6}}$with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-$\mathfrak{q}$strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-$\mathfrak{q}$and various enumeration strategies.

In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$-order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

Lattice sieving is asymptotically the fastest approach for solving the shortest vector problem (SVP) on Euclidean lattices. All known sieving algorithms for solving the SVP require space which (heuristically) grows as$2^{0.2075n+o(n)}$, where$n$is the lattice dimension. In high dimensions, the memory requirement becomes a limiting factor for running these algorithms, making them uncompetitive with enumeration algorithms, despite their superior asymptotic time complexity.We generalize sieving algorithms to solve SVP with less memory. We consider reductions of tuples of vectors rather than pairs of vectors as existing sieve algorithms do. For triples, we estimate that the space requirement scales as$2^{0.1887n+o(n)}$. The naive algorithm for this triple sieve runs in time$2^{0.5661n+o(n)}$. With appropriate filtering of pairs, we reduce the time complexity to$2^{0.4812n+o(n)}$while keeping the same space complexity. We further analyze the effects of using larger tuples for reduction, and conjecture how this provides a continuous trade-off between the memory-intensive sieving and the asymptotically slower enumeration.

We prove that every integer$n\geqslant 10$such that$n\not \equiv 1\text{ mod }4$can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author’s numerical computation regarding the generalised Riemann hypothesis to extend the explicit bounds of Ramaré–Rumely.

Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for$\mathbb{F}_{q}(t)$, excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus$q$and degree$n$number field$K$, generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod$q$of a certain fractional ideal${\mathcal{O}}_{K}^{\vee }\subset K$called the codifferent or ‘dual’, rather than from the ring of integers${\mathcal{O}}_{K}$itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$with$\unicode[STIX]{x1D6E5}_{K}$the discriminant of$K$. As a main result, we provide, for any$\unicode[STIX]{x1D700}>0$, a family of number fields$K$for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$.

For a$t$-nomial$f(x)=\sum _{i=1}^{t}c_{i}x^{a_{i}}\in \mathbb{F}_{q}[x]$, we show that the number of distinct, nonzero roots of$f$is bounded above by$2(q-1)^{1-\unicode[STIX]{x1D700}}C^{\unicode[STIX]{x1D700}}$, where$\unicode[STIX]{x1D700}=1/(t-1)$and$C$is the size of the largest coset in$\mathbb{F}_{q}^{\ast }$on which$f$vanishes completely. Additionally, we describe a number-theoretic parameter depending only on$q$and the exponents$a_{i}$which provides a general and easily computable upper bound for $C$. We thus obtain a strict improvement over an earlier bound of Canettiet al. which is related to the uniformity of the Diffie–Hellman distribution. Finally, we conjecture that$t$-nomials over prime fields have only$O(t\log p)$roots in$\mathbb{F}_{p}^{\ast }$when$C=1$.

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$, ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$, ordered by height, in which we compute ranks and $2$-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.