Symmetry of graphs and perfect state transfer in Grover walks

We study relationships between symmetry of graphs and perfect state transfer in Grover walks. Symmetry of graphs mathematically refers to automorphisms of graphs. When perfect state transfer occurs between two vertices, the following two statements hold. One is that automorphisms preserve the occurrence of perfect state transfer. The other is that the stabilizer subgroups of the automorphism groups with respect to those two vertices coincide. Using these results, we completely characterize circulant graphs up to valency $4$ that admit perfect state transfer. Its proof uses also algebraic number theory.Comment: 23 page

Finding Short Vectors in Structured Lattices with Reduced Quantum Resources

Leading protocols of post-quantum cryptosystems are based on the mathematical problem of finding short vectors in structured lattices. It is assumed that the structure of these lattices does not give an advantage for quantum and classical algorithms attempting to find short vectors. In this work we focus on cyclic and nega-cyclic lattices and give a quantum algorithmic framework of how to exploit the symmetries underlying these lattices. This framework leads to a significant saving in the quantum resources (e.g. qubits count and circuit depth) required for implementing a quantum algorithm attempting to find short vectors. We benchmark the proposed framework with the variational quantum eigensolver, and show that it leads to better results while reducing the qubits count and the circuit depth. The framework is also applicable to classical algorithms aimed at finding short vectors in structured lattices, and in this regard it could be seen as a quantum-inspired approach

On the arithmetic of generalized Fekete polynomials

For each prime number $p$ one can associate a Fekete polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this paper, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be a composite number. We then investigate the appearance of cyclotomic factors of these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.Comment: To appear in Experimental Mathematic

Fast norm computation in smooth-degree Abelian number fields

This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in $S$-unit searches (for, e.g., class-group computation) is computing a $\Theta(n\log n)$-bit norm of an element of weight $n^{1/2+o(1)}$ in a degree-$n$ field; this method then uses $n(\log n)^{3+o(1)}$ bit operations. An $n(\log n)^{O(1)}$ operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least $n^2(\log n)^{2+o(1)}$ bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm $n/(\log n)^{1+o(1)}$ times faster, and finish norm computations inside $S$-unit searches $n^2/(\log n)^{1+o(1)}$ times faster

A note on short invertible ring elements and applications to cyclotomic and trinomials number fields

Ring-SIS based $\Sigma$-protocols require a challenge set $\mathcal{C}$ in some ring $R$, usually an order in a number field $L$. These $\Sigma$-protocols impose various requirements on the subset $\mathcal{C}$, and finding a good, or even optimal, challenge set is a non-trivial task that involves making various trade-offs. Ring-SIS based $\Sigma$-protocols require a challenge set $\mathcal{C}$ in some ring $R$, usually an order in a number field $L$. These $\Sigma$-protocols impose various requirements on the subset $\mathcal{C}$, and finding a good, or even optimal, challenge set is a non-trivial task that involves making various trade-offs. In particular, (1) the set $\mathcal{C}$ should be `large', (2) elements in $\mathcal{C}$ should be `small', and (3) differences of distinct elements in $\mathcal{C}$ should be invertible modulo a rational prime $p$. Moreover, for efficiency purposes, it is desirable that (4) the prime $p$ is small, and that (5) it splits in many factors in the number field $L$. These requirements on $\mathcal{C}$ are subject to certain trade-offs, e.g., between the splitting behavior of the prime $p$ and its size. Lyubashevsky and Seiler (Eurocrypt 2018) have studied these trade-offs for subrings of cyclotomic number fields. Cyclotomic number fields possess convenient properties and as a result most Ring-SIS based protocols are defined over these specific fields. However, recent attacks have shown that, in certain protocols, these convenient properties can be exploited by adversaries, thereby weakening or even breaking the cryptographic protocols. In this work, we revisit the results of Lyubashevsky and Seiler and show that they follow from standard Galois theory, thereby simplifying their proofs. Subsequently, this approach leads to a natural generalization from cyclotomic to arbitrary number fields. We apply the generalized results to construct challenge sets in trinomial number fields of the form $\mathbb{Q}[X]/(f)$ with $f=X^n+aX^k+b \in \mathbb{Z}[X]$ irreducible. Along the way we prove a conjectured result on the practical applicability for cyclotomic number fields and prove the optimality of certain constructions. Finally, we find a new construction for challenge sets resulting in smaller prime sizes at the cost of slightly increasing the $\ell_2$-norm of the challenges