scholarly journals A Reduction of Integer Factorization to Modular Tetration

2020 ◽  
Vol 31 (04) ◽  
pp. 461-481
Author(s):  
Markus Hittmeir
Keyword(s):  
Exponential Function ◽  
Polynomial Time ◽  
Integer Factorization ◽  
Natural Numbers ◽  
Prime Factorization

Let [Formula: see text]. By [Formula: see text] and [Formula: see text], we denote the [Formula: see text] th iterate of the exponential function [Formula: see text] evaluated at [Formula: see text], also known as tetration. We demonstrate how an algorithm for evaluating tetration modulo natural numbers [Formula: see text] could be used to compute the prime factorization of [Formula: see text] and provide heuristic arguments for the efficiency of this reduction. Additionally, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.

scholarly journals Series Representation of Power Function

2018 ◽  
Author(s):  
Petro Kolosov
Keyword(s):  
Numerical Methods ◽  
Power Function ◽  
Exponential Function ◽  
Finite Differences ◽  
Series Representation ◽  
Discrete Case ◽  
Natural Numbers ◽  
Binomial Theorem ◽  
Binomial Expansion

In this paper described numerical expansion of natural-valued power function xn, in point x = x0 where n, x0 - natural numbers. Apply- ing numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared with solutions according to Newton’s Binomial theorem and MacMillan Double Bi- nomial sum. Additionally, in section 4 exponential function’s ex representation is shown.

scholarly journals A New LSB Attack on Special-Structured RSA Primes

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 838
Author(s):  
Amir Hamzah Abd Ghafar ◽  
Muhammad Rezal Kamel Ariffin ◽  
Muhammad Asyraf Asbullah
Keyword(s):  
Polynomial Time ◽  
Public Key Cryptosystem ◽  
Integer Factorization ◽  
Partial Knowledge ◽  
Rsa Cryptosystem ◽  
Factorization Problem ◽  
Integer Factorization Problem ◽  
Positive Integers ◽  
Vital Element ◽  
Great Danger

Asymmetric key cryptosystem is a vital element in securing our communication in cyberspace. It encrypts our transmitting data and authenticates the originality and integrity of the data. The Rivest–Shamir–Adleman (RSA) cryptosystem is highly regarded as one of the most deployed public-key cryptosystem today. Previous attacks on the cryptosystem focus on the effort to weaken the hardness of integer factorization problem, embedded in the RSA modulus, N = p q . The adversary used several assumptions to enable the attacks. For examples, p and q which satisfy Pollard’s weak primes structures and partial knowledge of least significant bits (LSBs) of p and q can cause N to be factored in polynomial time, thus breaking the security of RSA. In this paper, we heavily utilized both assumptions. First, we assume that p and q satisfy specific structures where p = a m + r p and q = b m + r q for a , b are positive integers and m is a positive even number. Second, we assume that the bits of r p and r q are the known LSBs of p and q respectively. In our analysis, we have successfully factored N in polynomial time using both assumptions. We also counted the number of primes that are affected by our attack. Based on the result, it may poses a great danger to the users of RSA if no countermeasure being developed to resist our attack.

Complexity, decidability and completeness

2006 ◽  
Vol 71 (2) ◽  
pp. 399-424 ◽  
Author(s):  
Douglas Cenzer ◽  
Jeffrey B. Remmel
Keyword(s):  
Polynomial Time ◽  
Predicate Logic ◽  
Completeness Theorem ◽  
Binary Representation ◽  
Polynomial Space ◽  
Natural Numbers ◽  
Decidable Theory ◽  
Consistent Extension ◽  
Propositional Theory ◽  
Exponential Space

AbstractWe give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

scholarly journals Implementation of Shor’s Quantum Factoring Algorithm using ProjectQ Framework

2019 ◽  
Vol 8 (6S3) ◽  
pp. 52-56
Keyword(s):  
Quantum Computing ◽  
Computer Science ◽  
Polynomial Time ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Prime Factorization ◽  
Shor's Algorithm ◽  
Computer Simulators ◽  
Testing And Evaluation ◽  
Factoring Algorithm

Prime number factorization is a problem in computer science where the solution to that problem takes super-polynomial time classically. Shor’s quantum factoring algorithm is able to solve the problem in polynomial time by harnessing the power of quantum computing. The implementation of the quantum algorithm itself is not detailed by Shor in his paper. In this paper, an approach and experiment to implement Shor’s quantum factoring algorithm are proposed. The implementation is done using Python and a quantum computer simulator from ProjectQ. The testing and evaluation are completed in two computers with different hardware specifications. User time of the implementation is measured in comparison with other quantum computer simulators: ProjectQ and Quantum Computing Playground. This comparison was done to show the performance of Shor’s algorithm when simulated using different hardware. There is a 33% improvement in the execution time (user time) between the two computers with the accuracy of prime factorization in this implementation is inversely proportional to the number of qubits used. Further improvements upon the program that has been developed for this paper is its accuracy in terms of finding the factors of a number and the number of qubits used, as previously mentioned.

scholarly journals Experimental Analysis of Attacks on RSA & Rabin Cryptosystems using Quantum Shor’s Algorithm

2021 ◽  
Author(s):  
Ritu Thombre ◽  
Babita Jajodia
Keyword(s):  
Communication Networks ◽  
Polynomial Time ◽  
Experimental Analysis ◽  
Data Security ◽  
Prime Numbers ◽  
Quantum Computers ◽  
Integer Factorization ◽  
Potential Threat ◽  
Shor's Algorithm ◽  
Cryptographic Algorithms

In this world of massive communication networks, data security and confidentiality are of crucial importance for maintaining secured private communication and protecting information against eavesdropping attacks. Existing cryptosystems provide data security and confidentiality by the use of encryption and signature algorithms for secured communication. Classical computers use cryptographic algorithms that use the product of two large prime numbers for generating public and private keys. These classical algorithms are based on the fact that integer factorization is a non-deterministic polynomial-time (NP) problem and requires super-polynomial time making it impossible for large enough integers. Shor’s algorithm is a well-known algorithm for factoring large integers in polynomial time and takes only O(b3) time and O(b) space on b-bit number inputs. Shor’s algorithm poses a potential threat to the current security system with the ongoing advancements of Quantum computers. This paper discusses how Shor’s algorithm will be able to break integer factorization-based cryptographic algorithms, for example, Rivest–Shamir–Adleman (RSA) and Rabin Algorithms. As a proof of concept, experimental analysis of Quantum Shor’s algorithm on existing public-key cryptosystems using IBM Quantum Experience is performed for factorizing integers of moderate length (seven bits) due to limitations of thirty-two qubits in present IBM quantum computers. In a nutshell, this work will demonstrate how Shor’s algorithm poses threat to confidentiality and authentication services.

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