Diffie-Hellman key-agreement and RSA cryptosystem are widely used to provide security in internet protocols. But both of the two algorithms are totally breakable using Shor's algorithms. This paper proposes two connected matrix-based key-agreements: (a) Diffie-Hellman Key-Agreement with Errors and (b) RSA-Resemble Key-agreement, which, respectively, bear resemblance to Diffie-Hellman key-agreement and RSA cryptosystem and thereby they gain some of the well-known security characteristics of these two algorithms, but without being subject to Shor's algorithms attacks. That is, the new schemes avoid the direct reliance on the hardness of Discrete Logarithm and Integer Factoring problems which are solvable by Shor's algorithms. The paper introduces a new family of quantum-safe hardness assumptions which consist of taking noisy powers of binary matrices. The new assumptions are derived from Decisional Diffie-Hellman (DDH) assumption in the general linear group GL(n,2) by introducing random noise into a quadruple similar to that which define the DDH assumption in GL(n,2(. Thereby we make certain that the resulting quadruple is secure against Shor's algorithm attack and any other DLP-based attack. Thence, the resulting assumptions, are used as basis for the two key-agreement schemes. We prove that these key-agreements are secure -- in key indistinguishability notion -- under the new assumptions.
Diffie-Hellman key-agreement and RSA cryptosystem are widely used to provide security in internet protocols. But both of the two algorithms are totally breakable using Shor's algorithms. This paper proposes two connected matrix-based key-agreements: (a) Diffie-Hellman Key-Agreement with Errors and (b) RSA-Resemble Key-agreement, which, respectively, bear resemblance to Diffie-Hellman key-agreement and RSA cryptosystem and thereby they gain some of the well-known security characteristics of these two algorithms, but without being subject to Shor's algorithms attacks. That is, the new schemes avoid the direct reliance on the hardness of Discrete Logarithm and Integer Factoring problems which are solvable by Shor's algorithms. The paper introduces a new family of quantum-safe hardness assumptions which consist of taking noisy powers of binary matrices. The new assumptions are derived from Decisional Diffie-Hellman (DDH) assumption in the general linear group GL(n,2) by introducing random noise into a quadruple similar to that which define the DDH assumption in GL(n,2(. Thereby we make certain that the resulting quadruple is secure against Shor's algorithm attack and any other DLP-based attack. Thence, the resulting assumptions, are used as basis for the two key-agreement schemes. We prove that these key-agreements are secure -- in key indistinguishability notion -- under the new assumptions.
The security of popular public key-cryptographic protocols, such as RSA, Diffie-Hellman key exchange and the Digital Signature Algorithm (DSA), is endangered by the advent of quantum computers. Shor brought a big breakthrough with his quantum algorithm that can be used to factor an arbitrarily large integer into the product of its prime factors, hence jeopardizing the security of RSA, and that at the same time also solves the Discrete Logarithm Problem, which raises issues for certain Diffie-Hellman based cryptosystems and digital signatures. It is hence crucial to upgrade our current tools for post-quantum cryptography: it should be infeasible, even using quantum algorithms, to break the new cryptosystems. Popular candidates include for example elliptic curve or lattice-based cryptography, but they share something in common: they are specific cases of the more general Hidden Subgroup and connected Hidden Shift Problem.
Today we witness an explosive growth in attention to Q-computing. Q-computing technologies, along with artificial intelligence (AI) and machine learning (ML) technologies, cloud and foggy computing, as well as technologies for collecting and streaming processing of Big Data and ETL, are constantly leading the lists of ""end-to-end"" information technologies for the digital economy of technologically developed countries of the world. One of the main reasons for this is the potential ability of quantum computers to solve some computational problems more efficiently than any of the most modern classical computers of the von Neumann architecture (supercomputers). The most expressive and interesting, from an applied point of view, examples of such problems are integer factorization, effectively performed by Shor's quantum algorithm, as well as record search in an unordered database, effectively solved by Grover's algorithm. This monograph contains the best practice for solving problems of quantum cryptanalysis to improve cyber security and resilience of the digital economy. The book discusses well-known and author's software implementations of promising quantum Shor algorithms, Grover, Simon et al. Shor's algorithm provides exponential acceleration of solving factorization problems, discrete logarithm problems (DLPs) and elliptic curve discrete logarithm problems (ECDLPs). The mentioned tasks are widely used in TLS, SSH or IPsec cryptographic applications of Internet/Intranet and IIoT/IoT networks, communication protocols based on Diffie–Hellman key agreements (dependent on the strength of the DLP or ECDLP), digital signature algorithms (DSA, ECDSA, RSA-PSS), public key encryption algorithms (El Gamal, RSA-OAEP), etc. In other words, Shor's quantum algorithm is potentially capable of violating these algorithms, and with them all the mechanisms of public-key cryptography deployed in cyberspace.
We survey the computational assumptions of various cryptographic schemes, and discuss the security threat posed by Shor's quantum algorithm. One-way functions form the basis of public-key cryptography. Although we have candidate hard problems that are believed to be one-way, none has been proven to be so. Therefore the security of the corresponding cryptographic schemes depends on the intractability assumptions of these problems. Two major species of such problems, factoring and discrete logarithm, are widely believed to be intractable, and serve as the basis of many popular schemes. However, these two problems turned out to be polynomial-time solvable on a hypothetical quantum computer using Shor's algorithm. This is the most worrisome long-term threat to current public-key cryptosystems. In the thesis we provide a review of existing cryptosystems, with a focus on their underlying computational assumptions and the security. Other than factoring and discrete logarithm, schemes have been proposed based on error-correcting codes, subset-sum and subset-product problems, lattice, polynomials, combinatorial group theory, number fields, etc. Many are to be furtherly evaluated in future research.
The emergence of some quantum algorithms has brought a serious threat to modern cryptography,among which Shor’s algorithm is the most important threatening algorithm for cryptanalysis currently.Shor’s algorithm can solve the integer factorization problem (IFP) and discrete logarithm problem (DLP) in polynomial-time,which makes the current widely used RSA,ElGamal and ECC public key cryptosystem unsafe any more under the quantum computing environment.Therefore,it is necessary to research the cryptanalysis in the quantum computing environment.Solving the IFP is the core idea of Shor’s algorithm for attacking RSA,but breaking RSA does not have to be solved by solving the IFP.A quantum algorithm is designed to attack the RSA cryptosystem starting from the angle of non-factorization.Focusing on the characteristics of RSA public key cryptosystem,using the quantum Fourier transform,the RSA plaintext M can be got by calculating the eth root modulus n .That is,without solving the IFP,RSA is broken.Different from the previous practices that cryptanalysts try to recover the private-key,a ciphertext-only attack algorithm for RSA,directly from recovering the plaintext M to start, is presented.Results show that the probability of success of the new algorithm is higher than that of Shor’s algorithm attacking RSA.At the same time,the new algorithm does not recover the RSA plaintext from the ciphertext without factoring the modulus n ,and avoids the restriction that the order of ciphertext C modules n is even.
Many cryptographic algorithms depend on computational complexity assumptions. Notorious cases are the RSA algorithm for public key criptography or the Diffie-Hellman key exchange protocol, to publicly agree on a common secret key. Both algorithms are known to be broken by quantum computing as well as those that can be reduced to a discrete logarithm problem. These are key algorithms in our digital society and are at the basis of everyday tasks, specially those that rely on digital signatures. The RSA algorithm, in particular, is probably the most used algorithm and is its assumed security the one that guards most of the e-commerce nowadays. In this case, it is the time complexity of finding the prime factors of a large number, that grows worse than polinomially with the size of the number, the main guardian of our cyberinfrastructure. The fact that a quantum computer can solve this problem in polynomial time using Shor's algorithm is seen as a potentially major disruption and has prompted security agencies to advice the progressive deprecation of these algorithms.
Shor,Boneh,Liptonon et al. present some remarkable quantum algorithms which can solve integer factoring problem, discrete logarithm problem and discrete logarithm problem on elliptic curves in quantum polynomial time. These quantum algorithms are great challenges to classical public key cryptographies based on the above-described hard problems. It seems that braid group is a kind of considerable public key cryptography platform, but current key agreement protocols, such as AAG,AAFG and BDH, all have different degrees of security weaknesses.This paper takes advantage of random braids and non-conjugate transformations to present an improved braid key agreement protocol related to AAG and AAFG, which can make the two communication parties securely share a common key over any insecure channel. This protocol can resists current length-based attacks, linear representation attacks and other conjugacy search attacks.