Miller-Rabin Research Articles

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The twin primes conjecture states that there are infinitely many twin primes. While studying this hypothesis, many important results were obtained, but the problem remains unsolved. In this work, the problem is studied from the side of experimental mathematics. Using the probabilistic Miller–Rabin primality test and parallel computing technologies, the distribution of prime pairs in the intervals (2n; 2n+1] is studied experimentally.

Variations on a Miller–Rabin Theme

The Miller–Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography.

An Intelligent Choice of Witnesses in the Miller–Rabin Primality Test. Reinforcement Learning Approach

An Intelligent Choice of Witnesses in the Miller–Rabin Primality Test. Reinforcement Learning Approach

Estimating the Distribution of Witnesses of the Primality of the Miller-Rabin Test

This article investigates the error distribution of the Miller-Rabin test for the class of tripleprime numbers. At first the current results on the class of semiprimes are presented. Further, a theoretical estimation of the average frequency for triple prime numbers on an interval is derived, and a comparative analysis with a practical result is demonstrated. Graphs and intermediate conclusions accompany all comparisons. A conclusion is also made about a possible direction for improving this estimation.

How to detect whether Shor’s algorithm succeeds against large integers without a quantum computer

Shor’s algorithm is a well-known probabilistic method for factoring large composite integers in polynomial-time on a quantum computer. The method computes the order r of a random element x in the group Z∗N and uses that information for splitting N with an application of the greatest common divisor algorithm. However, being probabilistic, the success of Shor’s algorithm relies on some special properties of N. If r is even and xr/2 £ -1 mod N, then gcd(xr/2 - 1, N) reveals a nontrivial factor of N and the method succeeds. But even assuming that r is even and being given the complete prime factorization of N it is not obvious whether xr/2 £ -1 mod N and, therefore, it is not easy to assert whether Shor’s algorithm would split N without running it and looking at its answer. We present a strategy for detecting whether the splitting occurs without any need for running the quantum order-finding algorithm, but we must be given the prime factorization of N. This has allowed us to produce the first direct evidence of the probability of success of Shor’s method. The composites chosen were the product of two randomly-generated probable primes of similar sizes that pass the Miller-Rabin test.

A New Probabilistic Primality Test

In this paper, a new general probabilistic primality test is presented. The estimated efficiency of the test turns out to be inferior to that of the Miller–Rabin test. However, some heuristic arguments indicate that the estimation of its efficiency is quite rough. This allows one to expect that the actual efficiency of the test is much higher.

On the Number of Witnesses in the Miller–Rabin Primality Test

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

A Practical Collision-Based Power Analysis on RSA Prime Generation and Its Countermeasure

We analyze the security of RSA prime generation implemented on embedded devices by a practical power analysis attack. Unlike previous differential power analysis-based attack on primality tests of RSA prime generation exploiting the deterministic relationship among multiple prime candidates manipulated by consecutive primality tests, we propose a collision-based power analysis attack on the Miller-Rabin test for a single prime candidate which can recover the secret prime with a single attempt by exploiting collision characteristics of simple power analysis resistant modular exponentiation algorithms. Hence, our attack does not require the incremental prime search assumption and is applicable when countermeasures against previous attacks are deployed since it also does not require the assumption of trial divisions with small primes on prime candidates. For a realistic setting, where five 512-bit modular exponentiations are operated on an ARM Cortex-M4 microcontroller as recommended by FIPS 186-4 standard, we successfully recover the secret exponent to an extent that a feasible exhaustive search is needed for the full recovery of the secret prime. This is a first practical result of recovering a full secret of modular exponentiation which manipulates 512-bit RSA primitives with collision-based power analysis in a single attempt, where the previous attack demonstrates the result for 192-bit ECC primitive implementations. We also present a countermeasure against our attack which requires only one additional modular subtraction for the loop of square-and-multiply-always exponentiation algorithm. An experimental result for the effectiveness of our proposed countermeasure is presented.

The Error Probability of the Miller–Rabin Primality Test

In our paper we give theoretical and practical estimations of the error probability in the well-known Miller–Rabin probabilistic primality test. We show that a theoretical probability of error 0.25 for a single round of the test is very overestimated and, in fact, error is diminishing with the growth of length of numbers involved by a rate limited with ln n/ $$\sqrt n $$ .

Pseudo-prime number simulation and its application for security purpose

Many public cryptography schemes rely on the use of prime numbers like for encryption and decryption. A prime number is one number that is widely used and large and consists of hundreds of digits, so it takes time to test whether the numbers are prime or not. Miller-Rabin is one algorithm that could be used to test prime number. Simulation to show how to test non-prime number elimination process can be used to determine the workings of the Miller-Rabin algorithm and also could be used as a media learning for students and lecturers to know how prime number test and generation.

Primes in quadratic unique factorization domains

The present paper is devoted to extension of a number of well-known results on natural primes for prime elements in quadratic UFD. We obtain analogues of Miller's, Euler's, Lucas' and Pocklington's criterions of primality in quadratic UFD. There is proved that an analogue of the Miller–Rabin test can be realized in quadratic UFD and extended the Rabin result on probability of successful work of the Miller–Rabin test. We construct RSA-cryptosystem in quadratic domains and prove that there hold similar properties to RSA-cryptosystem on integers.

An RSA Scheme based on Improved AKS Primality Testing Algorithm

In applied cryptography, RSA is a typical asymmetric algorithm, which is used in electronic transaction and many other security scenarios. RSA needs to generate large random primes. Currently, primality test mostly depends on probabilistic algorithms, such as the Miller-Rabin primality testing algorithm. In 2002, Agrawal et al. published the Agrawal–Kayal–Saxena (AKS) primality testing algorithm, which is the first generic, polynomial, deterministic and non-hypothetical algorithm for primality test. This paper proves the necessary and sufficient condition for AKS primality test. An improved AKS algorithm is proposed using Fermat’s Little Theorem. The improved algorithm becomes an enhanced Miller-Rabin probabilistic algorithm, which can generate primes as fast as the Miller-Rabin algorithm does.

Strong pseudoprimes, covering systems of congruences and generalized bent functions

In this paper, we consider three problems, which are the computation of strong pseudoprimes, covering systems of congruences and generalized bent functions. Our highlights include: (1) through programming, proving that 3 825 123 056 546 413 051 is the smallest composite passing Miller-Rabin test to the first nine prime bases; (2) proving Kims conjecture that exact covering systems of congruences in any algebraic number field must have repeated moduli; (3) proving that two classes of generalized bent functions do not exist.

A primality test for 𝐾𝑝ⁿ+1 numbers

In this paper we generalize the classical Proth’s theorem and the Miller-Rabin test for integers of the form N = K p n + 1 N=Kp^n+1 . For these families, we present variations on the classical Pocklington’s results and, in particular, a primality test whose computational complexity is O ~ ( log 2 ⁡ N ) \widetilde {O}(\log ^2 N) and, what is more important, that requires only one modular exponentiation modulo N N similar to that of Fermat’s test.

Quantum computation of prime number functions

We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

Quantum computation of prime number functions

We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

On practical aspects of the Miller-Rabin Primality Test

The well-known Miller-Rabin Primality Test MRPT is used to check naturals to be prime or composite. We study the dependence between the length of testing numbers and the number of rounds of MRPT sufficient to give the correct answer and give some recommendations how to choose a suitable set of bases at which MRPT runs more efficiently. Concluding the paper we state a number of theoretical problems which decision allows to improve an implementation of MRPT.

Acceleration Detection of Large (Probably) Prime Numbers

In order to avoid unnecessary applications of Miller-Rabin algorithm to the number in question, we resort to trial division by a few initial prime numbers, since such a division take less time. How far we should go with such a division is the that we are trying to answer in this paper?For the theory of the matter is fully resolved. However, that in practice we do not have much use. Therefore, we present a solution that is probably irrelevant to theorists, but it is very useful to people who have spent many nights to produce large (probably) prime numbers using its own software.

COMPARISON STUDY OF FERMAT, SOLOVAY-STRASSEN AND MILLER-RABIN PRIMALITY TEST USING MATHEMATICA 6.0

This paper presents three primality tests; Fermat test, Solovay-Strassen test, and Rabin-Miller test. Mathematica software is used to carry out the primality tests. The application of Fermat’s Litle Theorem as well as Euler’s Theorem on the tests was also discussed and this leads to the concept of pseudoprime. This paper is also discussed some results on pseudoprimes with certain range and do quantitative comparison. Those primality tests need to be evaluated in terms of its ability to compute as well as correctness in determining primality of given numbers. The answer to this is to create a source codes for those tests and evaluate them by using Mathematica 6.0. Those are Miller-Rabin test, Solovay-Strassen test, Fermat test and Lucas-Lehmer test. Each test was coded using an algorithm derived from number theoretic theorems and coded using the Mathematica version 6.0. Miller-Rabin test, SolovayStrassen test, and Fermat test are probabilistic tests since they cannot certainly identify the given number is prime, sometimes they fail. Using Mathematica 6.0, comparison study of primality test has been made and given the Miller- Rabin test as the most powerful test than other.

Number theory: a lively introduction with proofs, applications, and stories

Preface. To the Student. To the Instructor. Acknowledgements. 0. Prologue. 1. Numbers, Rational and Irrational. (Historical figures: Pythagoras and Hypatia). 1.1 Numbers and the Greeks. 1.2 Numbers you know. 1.3 A First Look at Proofs. 1.4 Irrationality of he square root of 2. 1.5 Using Quantifiers. 2. Mathematical Induction. (Historical figure: Noether). 2.1.The Principle of Mathematical Induction. 2.2 Strong Induction and the Well Ordering Principle. 2.3 The Fibonacci Sequence and the Golden Ratio. 2.4 The Legend of the Golden Ratio. 3. Divisibility and Primes. (Historical figure: Eratosthenes). 3.1 Basic Properties of Divisibility. 3.2 Prime and Composite Numbers. 3.3 Patterns in the Primes. 3.4 Common Divisors and Common Multiples. 3.5 The Division Theorem. 3.6 Applications of gcd and lcm. 4.The Euclidean Algorithm. (Historical figure: Euclid). 4.1 The Euclidean Algorithm. 4.2 Finding the Greatest Common Divisor. 4.3 A Greeker Argument that the square root of 2 isIrrational. 5. Linear Diophantine Equations. (Historical figure: Diophantus). 5.1 The Equation aX + bY = 1. 5.2 Using the Euclidean Algorithm to Find a Solution. 5.3 The Diophantine Equation aX + bY = n. 5.4 Finding All Solutions to a Linear Diophantine Equation. 6. The Fundamental Theorem of Arithmetic. (Historical figure: Mersenne). 6.1 The Fundamental Theorem. 6.2 Consequences of the Fundamental Theorem. 7. Modular Arithmetic. (Historical figure: Gauss). 7.1 Congruence modulo n. 7.2 Arithmetic with Congruences. 7.3 Check Digit Schemes. 7.4 The Chinese Remainder Theorem. 7.5 The Gregorian Calendar. 7.6 The Mayan Calendar. 8. Modular Number Systems. (Historical figure: Turing). 8.1 The Number System Zn: an InformalView. 8.2 The Number System Zn: Definitionand Basic Properties. 8.3 Multiplicative Inverses in Zn. 8.4 Elementary Cryptography. 8.5 Encryption Using Modular Multiplication. 9. Exponents Modulo n. (Historical figure: Fermat). 9.1 Fermat's Little Theorem. 9.2 Reduced Residues and the Euler \phi-function. 9.3 Euler's Theorem. 9.4 Exponentiation Ciphers with a Prime modulus. 9.5 The RSA Encryption Algorithm. 10. Primitive Roots. (Historical figure: Lagrange). 10.1 Zn. 10.2 Solving Polynomial Equations inZn. 10.3 Primitive Roots. 10.4 Applications of Primitive Roots. 11. Quadratic Residues. (Historical figure: Eisenstein) 11.1 Squares Modulo n 11.2 Euler's Identity and the Quadratic Character of -1 11.3 The Law of Quadratic Reciprocity 11.4 Gauss's Lemma 11.5 Quadratic Residues and Lattice Points. 11.6 The Proof of Quadratic Reciprocity. 12. Primality Testing. (Historical figure: Erdos). 12.1 Primality testing. 12.2 Continued Consideration of Charmichael Numbers. 12.3 The Miller-Rabin Primality test. 12.4 Two Special Polynomial Equations inZp. 12.5 Proof that Millar-Rabin is Effective. 12.6 Prime Certificates. 12.7 The AKS Deterministic Primality Test. 13. Gaussian Integers. (Historical figure: Euler). 13.1 Definition of Gaussian Integers 13.2 Divisibility and Primes in Z[i]. 13.3 The Division Theorem for the Gaussian Integers. 13.4 Unique Factorization in Z[i]. 13.5 Gaussian Primes. 13.6 Fermat's Two Squares Theorem. 14. Continued Fractions. (Historical figure: Ramanujan). 14.1 Expressing Rational Numbers as Continued Fractions. 14.2 Expressing Irrational Numbers as Continued Fractions. 14.3 Approximating Irrational Numbers Using ContinuedFractions. 14.4 Proving that Convergents are Fantastic Approximations. 15. Some Nonlinear Diophantine Equations. (Historical figure: Germain). 15.1 Pell's Equation 15.2 Fermat's Last Theorem 15.3 Proof of Fermat's Last Theorem for n = 4. 15.4 Germain's Contributions to Fermat's Last Theorem 15.5 A Geometric look at the Equation x4 +y4 = z2. Appendix: Axioms of Number Theory. A.1 What is a Number System? A.2 Order Properties of the Integers. A.3 Building Results From Our Axioms. A.4 The Principle of Mathematical Induction.