相关论文: On the Design and Optimization of a Quantum Polyno…

Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a "quantum attack" of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation…

Improving over an earlier construction by Kaye and Zalka, Maslov et al. describe an implementation of Shor's algorithm which can solve the discrete logarithm problem on binary elliptic curves in quadratic depth O(n^2). In this paper we show…

Elliptic curves over finite fields GF(2^n) play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this…

In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These…

Multiplication over binary fields is a crucial operation in quantum algorithms designed to solve the discrete logarithm problem for elliptic curve defined over $GF(2^n)$. In this paper, we present an algorithm for constructing quantum…

In previous research, quantum resources were concretely estimated for solving Elliptic Curve Discrete Logarithm Problem(ECDLP). In [1], the quantum algorithm was optimized for the binary elliptic curves and the main optimization target was…

We perform logical and physical resource estimation for computing binary elliptic curve discrete logarithms using Shor's algorithm on fault-tolerant quantum computers. We adopt a windowed approach to design our circuit implementation of the…

We show in some detail how to implement Shor's efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further…

We present an algorithm that, on input of a positive integer N together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N. Although it is unproved that this can be done for…

We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm…

A new algorithms for computing discrete logarithms on elliptic curves defined over finite fields is suggested. It is based on a new method to find zeroes of summation polynomials. In binary elliptic curves one is to solve a cubic system of…

We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuit $C$ acting on $n$ qubits, and any bit string $x\in\{0,1\}^n$, can compute the quantity $|< x |C|0^{\otimes n}>|^2$ to within any…

We present improved quantum circuits for elliptic curve scalar multiplication, the most costly component in Shor's algorithm to compute discrete logarithms in elliptic curve groups. We optimize low-level components such as reversible…

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially…

The elliptic curve discrete logarithm problem is of fundamental importance in public-key cryptography. It is in use for a long time. Moreover, it is an interesting challenge in computational mathematics. Its solution is supposed to provide…

Shor's algorithm is well-known for its capability to address the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. The enhancement of its quantum resources continues to be a crucial focus of research. Nevertheless, the…

We present a new method for solving the hidden polynomial graph problem (HPGP) which is a special case of the hidden polynomial problem (HPP). The new approach yields an efficient quantum algorithm for the bivariate HPGP even when the input…

The quantum algorithm with polynomial time for discrete logarithm problem proposed by Shor is one of the most significant quantum algorithms, but a large number of qubits may be required in the Noisy Intermediate-scale Quantum (NISQ) era.…

We present a quantum algorithm for multiplying two $n$-bit integers with overall circuit depth and $T$-depth both bounded by $O(\log^{2} n)$, while using $O(n^{2})$ gates and ancillary qubits. Our construction generates partial products via…

Since the elliptic curve discrete logarithms problem (ECDLP) was proposed, it has been widely used in cryptosystem because of its strong security. Although the proposal of the extended Shor's algorithm offers hope for cracking ECDLP, it is…