The Discrete Logarithm Problem over Prime Fields can be transformed to a Linear Multivariable Chinese Remainder Theorem (original) (raw)
The General Chinese Remainder Theorem
The American Mathematical Monthly, 1952
The Chinese remainder theorem deals with systems of modular equations. The classical variant requires the modules to be pairwise coprime. In this paper we discuss the general variant, which does not require this restriction on modules. We have selected and implemented several algorithms for the general Chinese remainder theorem. Moreover, we point out some interesting applications of this variant in secret sharing and threshold cryptography.
Solving selected problems on the Chinese remainder theorem
Annales Mathematicae et Informaticae, 2022
The Chinese remainder theorem provides the solvability conditions for the system of linear congruences. In section 2 we present the construction of the solution of such a system. Focusing on the Chinese remainder theorem usage in the field of number theory, we looked for some problems. The main contribution is in section 3, consisting of Problems 3.1, 3.2 and 3.3 from number theory leading to the Chinese remainder theorem. Finally, we present a different view of the solution of the system of linear congruences by its geometric interpretation, applying lattice points.
On Some Algebraic Properties of the Chinese Remainder Theorem with Applications to Real Life
Journal of Applied Mathematics and Computation, 2021
The study sought to establish some algebraic properties of the Chinese Remainder Theorem. The Chinese Remainder Theorem is an ancient but important mathematical theorem that enables one to solve simultaneous equations with respect to different modulo and makes it possible to reconstruct integers in a certain range from their residues modulo to the pairwise relatively prime modulo and also con-struct libraries for manipulations on very large integers. The study seeks to find out some real life applications of the Chinese Remainder Theorem in our everyday life activities especially in trading and in information security and retrieval avoiding any leakages to invaders or intruders. The study presented proofs of some theorems vital in the real life applications of the Chinese Remainder Theorem. In the study, we identified that in the statement of the Principal Ideal Domain and that of Rings can be classified as some algebraic properties of the Chinese Remainder Theorem
A generalization of the Chinese remainder theorem
2001
The Chinese Remainder Theorem is more than 2000 years old. About this, e can read in the [1]. The Theorem has been successfully applied in the algorithm recently developed for the calculus with the large numbers, and several elementary arithmetics problems are built onto this.
On the cubic sieve method for computing discrete logarithms over prime fields
International Journal of Computer Mathematics, 2005
In this paper, we report efficient implementations of the linear sieve and the cubic sieve methods for computing discrete logarithms over prime fields. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two times faster than the linear sieve method even in cases of small prime fields of the size about 150 bits. We also provide a heuristic estimate of the number of solutions of the congruence X 3 ≡ Y 2 Z (mod p) that is of central importance in the cubic sieve method.
Multivariable Chinese remainder theorem
Resonance, 2015
In this note we show a multivariable version of the Chinese remainder theorem: a system of linear modular equations ai1xi + ... + ainxn = bi mod mi, i = 1, ..., n has solutions if mi > 1 are pairwise relatively prime and in each row, at least one matrix element aij is relatively prime to mi. The solution x can be found in a parallelepiped of volume M = m1m2 • • • mn. The Chinese remainder theorem is the special case, where A has only one column and the parallelepiped has dimension 1 × 1 × ... × 1 × M .
A Survey of Discrete Logarithm Algorithms in Finite Fields
2019
The discrete logarithm in a finite group of large order has been widely applied in public key cryptosystem. In this paper, we investigate attempts to solve the discrete logarithm problem, leading towards finding the current computationally best algorithms for performing the backwards computation. Several proposed algorithms for computing discrete logarithms are known and we briefly discuss some of them. Furthermore, we present the most powerful general-purpose algorithm that is known today, called the index-calculus algorithm, and analyze its asymptotic performance. Finally, we discuss several technical issues that are important to the performance of the index-calculus algorithm, such as rapid methods to solve the systems of linear equations that arise in it.
On the reduction in multiplicative complexity achieved by the polynomial residue number system
IEEE Transactions on Signal Processing, 1992
The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.
The complexity of the Chinese Remainder Theorem
arXiv (Cornell University), 2023
The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class V generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless V is the class of semilattices, the problem is either coNP-complete or tractable.
RSA algorithm and the Chinese Remainder Problem
I Introduction This is written on the RSA algorithm and the Chinese Remainder Problem. II RSA algorithm and the Chinese Remainder Problem https://en.wikipedia.org/wiki/RSA\_(cryptosystem)#cite\_note-10 Operation The RSA algorithm involves four steps: key generation, key distribution, encryption and decryption. A basic principle behind RSA is the observation that it is practical to find three very large positive integers e, d and n such that with modular exponentiation for all integer m: and that even knowing e and n or even m it can be extremely difficult to find d. Additionally, for some operations it is convenient that the order of the two exponentiations can be changed and that this relation also implies: RSA involves a public key and a private key. The public key can be known by everyone and is used for encrypting messages. The intention is that messages encrypted with the public key can only be decrypted in a reasonable amount of time using the private key. The public key is represented by the integers n and e; and, the private key, by the integer d (although n is also used during the decryption process; so, it might be considered a part of the private key, too). m represents the message (previously prepared with certain technique explained below). The keys for the RSA algorithm are generated the following way: 1.Choose two distinct prime numbers p and q. • For security purposes, the integers p and q should be chosen at random, and should be similar in magnitude but 'differ in length by a few digits'[2] to make factoring harder. Prime integers can be efficiently found using a primality test.
AN APPROACH TO ELLIPTIC CURVES AND DISCRETE LOGARITHMIC PROBLEM.
This paper studies the mathematics of elliptic curves, starting with their derivation and the proof of how points upon them form an additive abelian group. I then worked on the mathematics necessary to use these groups for cryptographic purposes, specifically results for the group formed by an elliptic curve over a finite field, E(Fq). I examine the mathematics behind the group of torsion points, to which every point in E(Fq) belongs, and prove Hasse’s theorem along with a number of other useful results. I finish by describing how to define a discrete logarithmic problem using E(Fq) and showing how this can form public key cryptographic systems for use in both encryption and decryption key exchange.
2016
The hardness of discrete logarithm problem over finite fields is the foundation of many cryptographic protocols. The state-of-art algorithms for solving the corresponding problem are number field sieve, function field sieve and quasi-polynomial time algorithm when the characteristics of the finite field are medium to large, medium-small and small, respectively. There are mainly three steps in such algorithms: polynomial selection, factor base logarithms computation, and individual logarithm computation. Note that the former two steps can be precomputed for fixed finite field, and the database containing factor base logarithms can be used by the last step for many times. In certain application circumstances, such as Logjam attack, speeding up the individual logarithm step is vital. In this paper, we devise two methods to improve the individual logarithm step by exploring subfield structure when the extension degree n is composite. The first method applies to the case when the charact...
Elliptic divisibility sequences and the elliptic curve discrete logarithm problem
We use properties of the division polynomials of an elliptic curve E over a finite field Fq together with a pure result about elliptic divisibility sequences from the 1940s to construct a very simple alternative to the Menezes-Okamoto-Vanstone algorithm for solving the elliptic curve discrete logarithm problem in the case where #E(Fq) = q − 1.
Optimising the New Chinese Remainder Theorem 1 for the Moduli Set
International Journal of Computer Applications, 2016
This paper seeks to improve the performance of the New Chinese Remainder Theorem (CRT) using the new moduli set {2^(2n+2)+3,2^(2n+1)+1,2^2n+1,2}. This optimization is very important in order to minimize the cost of hardware implementation and to improve the reverse conversion speed. The major factor responsible for this high hardware cost and high reverse conversion time is the presence of multipliers in the hardware implementation of the reverse converters. This paper proposes the moduli set {2^(2n+2)+3,2^(2n+1)+1,2^2n+1,2}, which is applicable for applications requiring larger dynamic range. The moduli set must be relatively prime integers. The computation of multiplicative inverses can be eliminated. We employ the proposed moduli set to optimize the New CRT-I. This scheme can result in less memory and adder based reverse converters, which is shown to be better than known existing similar state of the art scheme.
Further results on Chinese remaindering
1997
We present an attack on the RSA cryptosystem in a case where the attacker has very few knowledge of the parameters: even the public modulus is unknown to him. Our attack is based on the presence of faults and on the use of the Chinese Remainder Theorem to perform computations. With this attack, a damaged smart card can partially be broken.
On the Complexity of Generalized Discrete Logarithm Problem
arXiv (Cornell University), 2022
Generalized Discrete Logarithm Problem (GDLP) is an extension of the Discrete Logarithm Problem. The goal is to find x ∈ Z s such g x mod s = y for a given g, y ∈ Z s. Generalized discrete logarithm is similar but instead of a single base element, uses a number of base elements which does not necessarily commute with each other. In this paper, we prove that GDLP is NP-hard for symmetric groups. Furthermore, we prove that GDLP remains NP-hard even when the base elements are permutations of at most 3 elements. Lastly, we discuss the implications and possible implications of our proofs in classical and quantum complexity theory.
2003
The problem of a computationally feasible method of finding the discrete logarithm in a (large) finite field is discussed, presenting the main algorithms in this direction. Some cryptographic schemes based on the discrete logarithm are presented. Finally, the theory of linear recurring sequences is outlined, in relation to its applications in cryptology.
Some applications of linear congruence from number theory
International Research Journal of Science, Technology, Education, and Management, 2023
In this research, two distinct areas of number theory and its use in computer science are combined. In this article, an investigation was conducted on the implementation of solutions to linear congruence problems. Linear congruence is a concept implying that two integers a and b are congruent modulo m (denoted as 𝑎𝑏 (𝑚𝑜𝑑 ≡𝑚)), if the difference between them is exactly proportional to 𝑚. The study is important in the field of number theory and computer science which brings many benefits and efficient solutions to various problems. Therefore, the study investigates the application of linear congruence through illustrative examples, to apply number theory in finding the ISBN number, in converting decimal numbers to binary, octal, and hexadecimal, and its application in encoding and decoding messages from the field of cryptography. This section of the paper can make it easier for mathematicians to apply problems involving linear congruences, especially for those who need basic expertise in number theory. The findings of the study show that the compatibility of the book ISBN format can be checked through linear congruence. Additionally, these findings demonstrate your understanding of how to convert decimal values to binary, octal, and hexadecimal using linear congruence in a fairly comprehensive manner. Additionally, because the idea of a linear congruence system is employed in the encoding and decoding of codes for network security and other purposes, the findings of this study may be helpful to researchers working in the field of cryptography.