arXiv subjects

Oğuz Yayla

Publications and source records attributed to Oğuz Yayla.

11 recordsLinked to original sources

arXiv2026

Quantum Circuit Realization and Grover Cryptanalysis of the Hybrid ARX-SPN Cipher GFSPX

The security of classical symmetric-key primitives is fundamentally challenged by the emergence of quantum computing, necessitating a rigorous evaluation of their post-quantum resilience. This paper presents a comprehensive quantum circuit realization and Grover cryptanalysis of GFSPX, a lightweight block cipher featuring a 64-bit data block and a 128-bit secret key. GFSPX utilizes a unique hybrid architecture that integrates a 4-branch generalized Feistel structure with both Addition-Rotation-XOR (ARX) and Substitution-Permutation Network (SPN) components. Our quantum implementation optimizes resource distribution by exploiting the inherent reversibility of the Feistel network and employing a compact ripple-carry adder for the ARX layers. The proposed architecture achieves a qubit-optimized footprint of 209 qubits with a baseline quantum cost of 32,498 and a circuit depth of 7,617. To evaluate the cipher's resistance against quantum adversaries, we construct a parallelized Grover oracle using three plaintext-ciphertext pairs to eliminate spurious matches. Our analysis reveals that the total quantum cost of a key-recovery attack on GFSPX is $1.12 \times 2^{159}$ quantum gates. Although this cost falls below the NIST Level 1 security threshold of $2^{170}$, the hybrid ARX-SPN design demonstrates a higher quantum attack resistance among other lightweight designs. These findings provide critical insights into the balance between classical efficiency and quantum resilience in next-generation cryptographic designs for resource-constrained environments.

quant-ph↗

arXiv2024

Families of sequences with good family complexity and cross-correlation measure

In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.

cs.IT↗

arXiv2023

Codes on Subgroups of Weighted Projective Tori

We obtain certain algebraic invariants relevant to study codes on subgroups of weighted projective tori inside an $n$-dimensional weighted projective space. As application, we compute all the main parameters of generalized toric codes on these subgroups of tori lying inside a weighted projective plane of the form $\Pp(1,1,a)$.

math.AG↗

arXiv2022

Butson-Hadamard matrices and Plotkin-optimal p^k-ary codes

A Butson-Hadamard matrix H is a square matrix of dimension n whose entries are complex roots of unity such that HH*= nI. In the first part of this work, some new results on generalized Gray map are studied. In the second part, codes obtained from Butson-Hadamard matrices and some bounds on the minimum distance of these codes are proved. In particular, we show that the code obtained from a Butson-Hadamard matrix meets the Plotkin bound under a non-homogeneous weight. We also give the parameters of some code families which are obtained from modified Butson-Hadamard matrices under a (non)homogeneous Gray map.

math.CO↗

arXiv2020

Partial direct product difference sets and sequences with ideal autocorrelation

In this paper, we study the sequences with (non-consecutive) two zero-symbols and ideal autocorrelation, which are also known as almost $m$-ary nearly perfect sequences. We show that these sequences are equivalent to $\ell$-partial direct product difference sets (PDPDS), then we extend known results on the sequences with two consecutive zero-symbols to non-consecutive case. Next, we study the notion of multipliers and orbit combination for $\ell$-PDPDS. Finally, we present a construction method for a family of almost quaternary sequences with ideal autocorrelation by using cyclotomic classes.

math.CO↗

arXiv2020

The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree $n$ over the finite field $\mathbb F_q$ where the coefficients of $x^{n-1}$ and $x$ vanish for $n\ge3$. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements $a\in\mathbb F_{q^n}$ for which Trace$(a)=0$ and Trace$(a^{-1})=0$. Besides, we apply the formula to give an upper bound on the number of distinct constructions of a family of sequences with good family complexity and cross-correlation measure.

math.NT↗

arXiv2020

Comparison of Randomized Solutions for Constrained Vehicle Routing Problem

In this short paper, we study the capacity-constrained vehicle routing problem (CVRP) and its solution by randomized Monte Carlo methods. For solving CVRP we use some pseudorandom number generators commonly used in practice. We use linear, multiple-recursive, inversive, and explicit inversive congruential generators and obtain random numbers from each to provide a route for CVRP. Then we compare the performance of pseudorandom number generators with respect to the total time the random route takes. We also constructed an open-source library github.com/iedmrc/binary-cws-mcs on solving CVRP by Monte-Carlo based heuristic methods.

math.OC↗

arXiv2018

Almost p-ary Sequences

In this paper we study almost $p$-ary sequences and their autocorrelation coefficients. We first study the number $\ell$ of distinct out-of-phase autocorrelation coefficients for an almost $p$-ary sequence of period $n+s$ with $s$ consecutive zero-symbols. We prove an upper bound and a lower bound on $\ell$. It is shown that $\ell$ can not be less than $\min\{s,p,n\}$. In particular, it is shown that a nearly perfect sequence with at least two consecutive zero symbols does not exist. Next we define a new difference set, partial direct product difference set (PDPDS), and we prove the connection between an almost $p$-ary nearly perfect sequence of type $(γ_1, γ_2)$ and period $n+2$ with two consecutive zero-symbols and a cyclic $(n+2,p,n,\frac{n-γ_2 - 2}{p}+γ_2,0,\frac{n-γ_1 -1}{p}+γ_1,\frac{n-γ_2 - 2}{p},\frac{n-γ_1 -1}{p})$ PDPDS for arbitrary integers $γ_1$ and $γ_2$. Then we prove a necessary condition on $γ_2$ for the existence of such sequences. In particular, we show that they don't exist for $γ_2 \leq -3$.

math.CO↗

arXiv2014

Nearly perfect sequences with arbitrary out-of-phase autocorrelation

In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a $p$-ary NPS of period $n$ and type $γ$ and a cyclic $(n,p,n,\frac{n-γ}{p}+γ,0,\frac{n-γ}{p})$-DPDS for an arbitrary integer $γ$. Next, we present the necessary conditions for the existence of a $p$-ary NPS of type $γ$. We apply this result for excluding the existence of some $p$-ary NPS of period $n$ and type $γ$ for $n \leq 100$ and $\vert γ\vert \leq 2$. We also prove the similar results for an almost $p$-ary NPS of type $γ$. Finally, we show the non-existence of some almost $p$-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.

math.CO↗

arXiv2014

Family complexity and cross-correlation measure for families of binary sequences

We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al.\ in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family $(e_{i,1},\ldots,e_{i,N})\in \{-1,+1\}^N$, $i=1,\ldots,F$, of binary sequences of length $N$ in terms of the cross-correlation measure of its dual family $(e_{1,n},\ldots,e_{F,n})\in \{-1,+1\}^F$, $n=1,\ldots,N$. We apply this result to the family of sequences of Legendre symbols with irreducible quadratic polynomials modulo $p$ with middle coefficient $0$, that is, $e_{i,n}=\left(\frac{n^2-bi^2}{p}\right)_{n=1}^{(p-1)/2}$ for $i=1,\ldots,(p-1)/2$, where $b$ is a quadratic nonresidue modulo $p$, showing that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order.

math.NT↗

arXiv2014

Non-Existence of Some Nearly Perfect Sequences, Near Butson-Hadamard Matrices, and Near Conference Matrices

In this paper we study the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matrices and (near) conference matrices. Firstly, we apply a result on vanishing sums of roots of unity and a result of Brock on the unsolvability of certain equations over a cyclotomic number field to derive non-existence results for near BH matrices and near conference matrices. Secondly, we refine the idea of Brock in the case of cyclotomic number fields whose ring of integers is not a principal ideal domains and get many new non-existence results.

math.NT↗