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New algorithm for the elliptic curve discrete logarithm problem with auxiliary inputs

APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2017年第2期28卷 99-108页

作者： Weng, Jiang Dou, Yunqi Ma, Chuangui Air Force Engn Univ Xian Peoples R China State Key Lab Math Engn & Adv Comp Zhengzhou Peoples R China Army Aviat Inst Basic Dept Beijing Peoples R China

The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group of order p and some elements , an attacker can recover in the case of with running time of group operations by using storage ( or 1 for case or case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points and multiplicative cyclic group are given, where d is a prime, is the order of K and is the Euler totient function, the secret key can be solved in group operations by using storage.

关键词： elliptic curve discrete logarithm problem Auxiliary inputs Baby-step giant-step Cheon's algorithm

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Design and implementation of an efficient fair off-line E-cash system based on elliptic curve discrete logarithm problem

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JOURNAL OF COMMUNICATIONS AND NETWORKS 2002年第2期4卷 81-89页

作者： Lee, M Ahn, G Kim, J Park, J Lee, B Kim, K Lee, H ICU Taejon South Korea

In this paper, we design and implement an efficient fair off-line electronic cash system based on elliptic curve discrete logarithm problem (ECDLP), in which the anonymity of coins is revocable by a trustee in case of dispute. To achieve this, we employ the Petersen and Poupard's electronic cash system [1] and extend it by using an elliptic curve over the finite field GF(2(n)). This naturally reduces message size by 85% compared with the original scheme and makes a smart card to store coins easily. Furthermore, we use the Baek et al. Is provably secure public key encryption scheme [2] to improve the security of electronic cash system. As an extension, we propose a method to add atomicity into new electronic cash system. To the best of our knowledge, this is the first result to implement a fair off-line electronic cash system based on ECDLP with provable security.

关键词： electronic cash anonymity revocation atomicity elliptic curve discrete logarithm problem

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A Las Vegas Algorithm to Solve the elliptic curve discrete logarithm problem 19th

A Las Vegas Algorithm to Solve the Elliptic Curve Discrete L...

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19th International Conference on Cryptology in India (INDOCRYPT)

作者： Mahalanobis, Ayan Mallick, Vivek Mohan Abdullah, Ansari IISER Pune Pune Maharashtra India Savitribai Phule Pune Univ Pune Maharashtra India

ISBN: (纸本)9783030053789;9783030053772

In this paper, we describe a new Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The algorithm depends on a property of the group of rational points on an elliptic curve and is thus not a generic algorithm. The algorithm that we describe has some similarities with the most powerful index-calculus algorithm for the discrete logarithm problem over a finite field. The algorithm has no restriction on the finite field over which the elliptic curve is defined.

关键词： elliptic curve discrete logarithm problem

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Quantum Cryptanalysis Landscape of Shor's Algorithm for elliptic curve discrete logarithm problem 22nd

Quantum Cryptanalysis Landscape of Shor's Algorithm for Elli...

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22nd World Conference on Information Security Application (WISA)

作者： Larasati, Harashta Tatimma Kim, Howon Pusan Natl Univ Sch Comp Sci & Engn Busan 609735 South Korea Inst Teknol Bandung Sch Elect Engn & Informat Bandung 40116 Indonesia

ISBN: (纸本)9783030894320;9783030894313

Shor's algorithm is recognized as one of the most influential algorithms that shape the research interest in quantum computation and quantum cryptanalysis today. In particular, the algorithm is one of the firsts to show clear applicability of quantum computation to solve classically intractable problems underpinning the widely used public-key cryptosystems (i.e., RSA and ECC), igniting an ever-growing research interest in the respective fields. However, of the two algorithms introduced by Peter Shor in 1994, i.e., Shor's quantum factoring algorithm and Shor's discrete logarithm problem, the latter is not discussed as often. In this paper, we discuss the works done on Shor's algorithm to solve the elliptic curve discrete logarithm problem. In particular, we introduce the methods and optimizations proposed by researchers from the early era to the most recent ones and map them to obtain a comprehensive overview of the existing works. From our mapping, we analyze some of the trends in optimizations that help carve the research direction in Shor's algorithm for the elliptic curve discrete logarithm problem.

关键词： Shor's algorithm elliptic curve discrete logarithm problem Quantum cryptanalysis Review

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AN O(m²)-DEPTH QUANTUM ALGORITHM FOR THE elliptic curve discrete logarithm problem OVER GF(2^m)^a

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QUANTUM INFORMATION & COMPUTATION 2009年第7-8期9卷 610-621页

作者： Maslov, Dmitri Mathew, Jimson Cheung, Donny Pradhan, Dhiraj K. Univ Waterloo Dept Phys & Astron Waterloo ON N2L 3G1 Canada Natl Sci Fdn Directorate Comp & Informat Sci & Engn Arlington VA 22230 USA Univ Bristol Dept Comp Sci Bristol BS8 1UB Avon England Univ Calgary Dept Comp Sci Calgary AB T2N 1N4 Canada

We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over GF(2(m)). We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation, executable in the Linear Nearest Neighbor (LNN) architecture, is O(m(2)), which is an improvement over the previous bound of O(m(3)) derived assuming no architectural restrictions.

关键词： Quantum Circuits Cryptography elliptic curve discrete logarithm problem Quantum Algorithms

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BASE OF EXPONENT REPRESENTATION MATTERS - MORE EFFICIENT REDUCTION OF discrete logarithm problem AND elliptic curve discrete logarithm problem TO THE QUBO problem

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QUANTUM INFORMATION & COMPUTATION 2024年第7-8期24卷 541-564页

作者： Wronski, Micha Dzierzkowski, Lukasz NASK Natl Res Inst Dept Cryptol Kolska 12 Str PL-01045 Warsaw Poland Mil Univ Technol Warsaw Fac Cybernet Kaliskiego 2 Str PL-00908 Warsaw Poland

This paper presents further improvements in the transformation of the discrete logarithm problem (DLP) and elliptic curve discrete logarithm problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution. The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field F-p to the QUBO problem may be obtained using approximately 1.89n(2) logical variables for n being the bitlength of prime p, instead of the 2n(2) which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field F-47 . Similarly, for the elliptic curve discrete logarithm problem over the prime field F-p , allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for n being the bitlength of prime p, from 3n(3) to 6n(3) /( )log(2)(3/ 4 n ) logical variables, in the case of Edwards curves.

关键词： elliptic curve discrete logarithm problem D-Wave quantum annealing crypt- analysis

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FAIL-STOP DESIGNATED RECIPIENT SIGNATURE SCHEME BASED ON elliptic curve discrete logarithm problem

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS

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ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2013年第31期 205-218页

作者： Tahat, Nedal Hashemite Univ Fac Sci Dept Math Zarqa 13133 Jordan

This paper describes a new digital signature called fail-stop designated recipient signature scheme based on elliptic curve discrete logarithm (ECDLP). The scheme allows a signer and an intended recipient to co-operatively provide a proof of forgery if an attacker can successfully forge a signature on a message m. The scheme also provides that the intended recipient is the only entity to verify the resulting signature and capable to prove the validity of signature to any third party. With this property, we show that our new signature scheme is the best alternative to solve certain problems concerning the protection of confidential documents especially those which are personally sensitive (to the owner) and is also applicable for group of recipients to verify a signature jointly (shared verification) in a group oriented environment. The scheme utilizes the inherent advantage of elliptic curve cryptosystem in terms of smaller key size and lower computational overhead to its counterpart public cryptosystems such as RSA and AlGamal. Compared with the Ismail and Yahya scheme [1], we show that our scheme is more efficient in terms of both computation and communication complexity.

关键词： Cryptography Digital signature elliptic curve discrete logarithm problem Fail-stop designated recipient signature and zero knowledge Protocol

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Convertible multi-authenticated encryption scheme with verification based on elliptic curve discrete logarithm problem

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INTERNATIONAL JOURNAL OF COMPUTER APPLICATIONS IN TECHNOLOGY 2016年第3期54卷 229-235页

作者： Tahat, Nedal Hashemite Univ Fac Sci Dept Math PO 150459 Zarqa 13115 Jordan

The work presents a Convertible Multi-Authenticated Encryption (CMAE) scheme based on the elliptic curve discrete logarithm problem (ECDLP). The proposed scheme intended to develop the CMAE scheme to improve performance as compared to the other existing schemes. Most CMAE schemes are very high in computational overhead and memory usage problem. This scheme desired to reduce time-consuming problem by using elliptic curve Cryptography (ECC). In addition, the proposed scheme provides good resistance to the chosen plain-text attacks and is proven to be secure. As for efficiency, the computation cost of the proposed scheme is smaller than the existing schemes, such as Lu et al.'s scheme.

关键词： multi-authenticated encryption elliptic curve discrete logarithm problem chosen plain-text attacks encryption

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Minimizing CNOT-count in quantum circuit of the extended Shor’s algorithm for ECDLP

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Cybersecurity 2025年第1期8卷 153-179页

作者： Xia Liu Huan Yang Li Yang State Key Laboratory of Information Security Institute of Information EngineeringChinese Academy of SciencesBeijingChina School of Cyber Security University of Chinese Academy of SciencesBeijingChina Beijing Youzhuju Network Technology Co. Ltd.BeijingChina

The elliptic curve discrete logarithm problem(ECDLP)is a popular choice for cryptosystems due to its high level of ***,with the advent of the extended Shor’s algorithm,there is concern that ECDLP may soon be *** the algorithm does ofer hope in solving ECDLP,it is still uncertain whether it can pose a real threat in *** the perspective of the quantum circuits of the algorithm,this paper analyzes the feasibility of cracking ECDLP using an ion trap quantum computer with improved quantum circuits for the extended Shor’s *** give precise quantum circuits for extended Shor’s algorithm to calculate discrete logarithms on elliptic curves over prime felds,including modular subtraction,three diferent modular multiplication,and modular ***,we incorporate and improve upon windowed arithmetic in the circuits to reduce the *** previous studies mostly focused on minimizing the number of qubits or the depth of the circuit,we focus on minimizing the number of CNOT gates in the circuit,which greatly afects the running time of the algorithm on an ion trap quantum ***,we begin by presenting implementations of basic arithmetic operations with the lowest known CNOT-counts,along with improved constructions for modular inverse,point addition,and windowed ***,we precisely estimate that,to execute the extended Shor’s algorithm with the improved circuits to factor an n-bit integer,the CNOT-count required is1237n^(3)/log n+2n^(2)+***,we analyze the running time and feasibility of the extended Shor’s algorithm on an ion trap quantum computer.

关键词： elliptic curve discrete logarithm problem Extended Shor’s algorithm Quantum circuits Ion trap quantum computer

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Remarks on elliptic curve discrete logarithm problems

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2000年第1期E83A卷 17-23页

作者： Kanayama, N Kobayashi, T Saito, T Uchiyama, S Waseda Univ Grad Sch Sci & Engn Tokyo 1690085 Japan NTT Labs Yokosuka Kanagawa 2390847 Japan Telecommun Adv Org Japan Yokohama Kanagawa 2210031 Japan

The MOV and FR algorithms, which are representative attacks on elliptic curve cryptosystems, reduce the elliptic curve discrete logarithm problem (ECDLP) to the discrete logarithm problem in a finite field. This paper studies these algorithms and introduces the following three results. First, we show an explicit condition under which the MOV algorithm can be applied to non-supersingular elliptic curves. Next, by comparing the effectiveness of the MOV algorithm to that of the FR algorithm, it is explicitly shown that the condition needed for the MOV algorithm to be subexponential is the same as that for the FR algorithm except for elliptic curves of trace two. Finally, a new explicit reduction algorithm is proposed for the ECDLP over elliptic curves of trace two. This algorithm differs from a simple realization of the FR algorithm. Furthermore, pie show, by experimental results, that the running time of the proposed algorithm is shorter than that of the original FR algorithm.

关键词： elliptic curve discrete logarithm problem MOV algorithm FR algorithm

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