Related papers: Trustless unknown-order groups
Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is $\tilde O(m^5)$. In this work we propose a modification of the algorithm for which the number of qubits…
Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are…
We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear…
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers,…
A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to…
We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by…
Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute…
Group twirling is crucial in quantum information processing, particularly in randomized benchmarking and random compiling. While protocols based on Pauli twirling have been effectively crafted to transform arbitrary noise channels into…
The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…
The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for…
We consider classes of fundamental groups of complements of various kinds of codimension 2 embeddings and show that, in general, the problem of deciding whether or not a group in one class belongs to a smaller class is algorithmically…
Polycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups.…
We revisit the selection problem, namely that of computing the $i$th order statistic of $n$ given elements, in particular the classic deterministic algorithm by grouping and partition due to Blum, Floyd, Pratt, Rivest, and Tarjan (1973).…
In addition to the evaluation of high-order loop contributions, the precision and predictive power of perturbative QCD (pQCD) predictions depends on two important issues: (1) how to achieve a reliable, convergent fixed-order series, and (2)…
Universal compression of patterns of sequences generated by independently identically distributed (i.i.d.) sources with unknown, possibly large, alphabets is investigated. A pattern is a sequence of indices that contains all consecutive…
Cohen and Lenstra detailed a heuristic for the distribution of odd p-class groups for imaginary quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary…
We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to…
This work considers a generalization of Grover's search problem, viz., to find any one element in a set of acceptable choices which constitute a fraction f of the total number of choices in an unsorted data base. An infinite family of…
Since the work of Jennings (1955), it is well-known that any finitely generated torsion-free nilpotent group can be embedded into unitriangular integer matrices $UT_N(Z)$ for some $N$. In 2006, Nickel proposed an algorithm to calculate such…
We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the Fractional Quantum Hall Effect state at Landau level filling fraction nu=5/2. Since the braid group representation…