Primality Testing Research Articles

Primality test and primes enumeration using odd numbers indexation

Odd numbers can be indexed by the map k(n)=(n-3)⁄2,n∈2N+3. We first propose a basic primality test using this index function that was first introduced in [8]. Input size of operations is reduced which improves computational time by a constant. We then apply similar techniques to Atkin’s prime-numbers sieve which uses modulus operations and finally to Pritchard’s wheel sieve, in both case yielding similar results.

Primality test with pair of Lucas sequences

Lucas sequences and their applications play vital role in the study of primality tests in number theory. There are several known tests for primality of positive integer $N$ using Lucas sequences which are based on factorization of $(N \pm 1)$ [2] [13]. In this paper we give a primality test for odd positive integer $N>1$ by using the set $L(\Delta, N)$ where $L(\Delta, N)$ is a set of $S(N)$ distinct pair of Lucas sequences $(V_n (a,1),U_n (a,1))$, where $S(N)$ for $N=p_1^{e_1}.p_2^{e_2}\ldots p_s^{e_s}$ is given as $S(N)=LCM[\{p_{i}^{e_{i}-1}(p_i-(\frac{\Delta}{p_i}))\}_{i=1}^s]$ and $\Delta=a^2-4$ for some fixed integer $a$.

A primality test for $$4Kp^n-1$$ numbers

We present a Lucasian type primality test, not explicitly based on Lucas sequences, for numbers written in the form $$N=4Kp^n-1$$. This test is a generalization of the classical Lucas–Lehmer test for Mersenne numbers using as underlying group $$\mathcal {G}_N:=\{z \in (\mathbb {Z}/N\mathbb {Z})[i]: z\overline{z} \equiv 1 \pmod N\}$$.

Primality Testing for Numbers of the Form h · 2n ± 1

This paper studies the problem of primality testing for numbers of the form h · 2n ± 1, where h < 2n is odd, and n is a positive integer. The authors describe a Lucasian primality test for these numbers in certain cases, which runs in deterministic quasi-quadratic time. In particular, the authors construct a Lucasian primality test for numbers of the form 3 · 5 · 17 · 2n ± 1, where n is a positive integer, in half of the cases among the congruences of n modulo 12, by means of a Lucasian sequence with a suitable seed not depending on n. The methods of Bosma (1993), Berrizbeitia and Berry (2004), Deng and Huang (2016) can not test the primality of these numbers.

Some Interesting Consequences of Furstenberg Topology

Metrizability of Furstenberg topology on integers gives some strange consequences. Besides a primality test for an integer to be a prime, it shows that primes are not dense in integers. It illustrates the construction of a series of positive numbers having a negative sum. Furthermore, it re-establishes the fact that the probability of a randomly chosen integer being prime is zero.

Analysis of File Security with Three-Pass Protocol Scheme Using Massey-Omura Algorithm In Android

Today’s data transactions using the android device is commonly done anytime. Cryptography is needed to secure data to be transmitted to ensure data security from the manipulation of others. To secure the data, one of cryptography scheme is Three-Pass Protocol, this scheme allows data transmission without key exchange, so this scheme is more secure and efficient. A Massey-Omura algorithm is an asymmetric key cryptography system that will be used in this research. For primality testing in key and prime number generator, the Agrawal-Biswas algorithm will be used. The strength of Massey-Omura is the difficulty of calculating enormous discrete logarithmic and Three-Pass Protocol with his no key exchange capabilities make this method more secure. In this research, the Mean Squared Error value obtained is 0 and the value of Peak Signal to Noise Ratio obtained is infinite. The complexity of the Massey-Omura algorithm and the Agrawal-Biswas algorithm is Θ (n). Graph of file extension against processing time shows a stable graph that means file extension is not affecting processing time. Graph of file size against processing time shows a linear graph that means a processing time would increase along with file size.

Optimized AKS Primality Testing: A Fluctuation Theory Perspective

The AKS algorithm is an important breakthrough in showing that primality testing of an integer can be done in polynomial time. In this paper, we study the optimization of its runtime. Namely, given a finite cardinality set of alphabets of a deterministic polynomial runtime Turing machine and the number of strings of an arbitrary input integer whose primality is to be tested as the system parameters, we consider the randomized AKS primality testing function as the objective function. Under randomization of the system parameters, we have shown that there are definite signatures of the local and global instabilities in the AKS algorithm. We observe that instabilities occur at the extreme limits of the parameters. It is worth mentioning that Fermat’s little theorem and Chinese remaindering help with the determination of the underlying stability domains. On the other hand, in the realm of the randomization theory, our study offers fluctuation theory structures of the AKS primality testing of an integer through its maximum number of irreducible factors. Finally, our optimization theory analysis anticipates a class of real-world applications for future research and developments, including optimal online security, system optimization and its performance improvements, (de)randomization techniques, and beyond, e.g., polynomial time primality testing, identity testing, machine learning, scientific computing, coding theory, and other stimulating optimization problems in a random environment.

The Comparison of Methods for Generating Prime Numbers between The Sieve of Eratosthenes, Atkins, and Sundaram

Prime numbers are unique numbers. Prime numbers are numbers that only have a dividing factor consisting of numbers 1 and the number itself. The prime numbers from 1 to n that are relatively small can be generated manually, but a prime number generator algorithm is needed to generate prime numbers on a large scale. This study compares three prime number generator algorithms, namely: The Sieve of Eratosthenes, The Sieve of Atkins, and The Sieve of Sundaram. These three sieve algorithms have their own differences in generating prime numbers. The Sieve of Eratosthenes uses a simpler method by crossing multiples of prime numbers and marking them as non-prime numbers. The Sieve of Atkins uses several requirements for quadratic equations and modulus in determining prime numbers. The Sieve of Sundaram has an algorithm similar to The Sieve of Atkins, but there are requirements for linear equations to determine prime numbers. This study aims to see a comparison of these three algorithms in terms of accuracy and speed in generating prime numbers on a large scale. The results of this study indicate The Sieve of Eratosthenes, Atkins and Sundaram algorithms can generate large numbers of prime numbers with good accuracy, this was tested by the Fermat Primality Test Algorithm. The conclusion that can be drawn from this study, The Sieve of Eratosthenes have a faster time to generate prime numbers on a large scale than the other two algorithms.

Review of Number Theory

There are probably at least as many different approaches to number theory as there are books written about it. Some broad distinctions include those taking an historical versus (say) a purely modern approach, with many gradations in between, or those that are algebraically oriented (e.g., with an emphasis on reciprocity laws, or questions that relate to algebraic geometry), or still others that are more analytic. The book under review is definitely in the latter category. The "message" of the book is in the title, as primes and their density are the principle concern. In accordance with that theme, a highlight of the book is a complete proof of the prime number theorem. However, the theme and its variations are taken as springboards to other important fields, including aspects of algebraic number theory, as well as applications, such as primality testing and cryptography.

Toward the Unification of Physics and Number Theory

This paper introduces the notion of simplex-integers and shows how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. A geometric analogue to the primality test is introduced: when [Formula: see text] is prime, it divides [Formula: see text] for all [Formula: see text]. The geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound [Formula: see text]-simplex or associated [Formula: see text] lattice is discovered, a deep understanding of the error factor of the prime number theorem would be realized — the error factor corresponding to the distribution of the non-trivial zeta zeros, which might be the mysterious link between physics and the Riemann hypothesis [D. Schumayer and D. A. W. Hutchinson, Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83 (2011) 307]. It suggests how quantum gravity and particle physicists might benefit from a simplex-integer-based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax.

Information Security Methods—Modern Research Directions

In Tomsk University of Control Systems and Radioelectronics (TUSUR) one of the main areas of research is information security. The work is carried out by a scientific group under the guidance of Professor Shelupanov. One of the directions is the development of a comprehensive approach to assessing the security of the information systems. This direction includes the construction of an information security threats model and a protection system model, which allow to compile a complete list of threats and methods of protection against them. The main directions of information security tools development are dynamic methods of biometrics, methods for generating prime numbers for data encryption, steganography, methods and means of data protection in Internet of Things (IoT) systems. The article presents the main results of research in the listed areas of information security. The resultant properties in symmetric cryptography are based on the properties of the power of the generating functions. The authors have obtained symmetric principles for the development of primality testing algorithms, as discussed in the Appendix.

Primality testing with Gaussian periods

We exhibit a deterministic algorithm that, for some effectively computable real number c , decides whether a given integer n &gt; 1 is prime within time \mathrm {log} n)^6\cdot(2+\mathrm {log\log} n)^c . The same result, with 21/2 in place of 6 , was proved by Agrawal, Kayal, and Saxena. Our algorithm follows the same pattern as theirs, performing computations in an auxiliary ring extension of \mathbb Z/n\mathbb Z . We allow our rings to be generated by Gaussian periods rather than by roots of unity, which leaves us greater freedom in the selection of the auxiliary parameters and enables us to obtain a better run time estimate. The proof depends on results in analytic number theory and on the following theorem from additive number theory, which was provided by D. Bleichenbacher: if t is a real number with 0 &lt; t \le1 , and S is an open subset of the interval (0,t) with \int_S\mathrm d x/x &gt; t , then each real number greater than or equal to 1 is in the additive semigroup generated by S . A byproduct of our main result is an improved algorithm for constructing finite fields of given characteristic and approximately given degree.

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.

Study on Deterministic and Probabilistic Computation of Primality Test

Every natural number greater than 1 is either prime or composite. Those numbers which have only two factors i.e., 1 and itself are considered as prime and rest are considered as composite numbers. The property of a number for being prime is called primality. In this survey paper we will deal with the algorithms that can check whether the number is prime or not i.e., primality test. This study is the detailed survey of probabilistic and deterministic algorithms like Fermat’s theorem of primality test, AKS theorem, Miller Rabin’s test, Solvay Strassen’s theorem etc. We will discuss different parameters regarding algorithms which are best for testing large prime numbers. Many aspects will be considered while discussing these algorithms.

Introducing Primality Testing Algorithm with an Implementation on 64 bits RSA Encryption Using Verilog

A new structure to develop 64-bit RSA encryption engine on FPGA is being presented in this paper that can be used as a standard device in the secured communication system. The RSA algorithm has three parts i.e. key generation, encryption and decryption. This procedure also requires random generation of prime numbers, therefore, we are proposing an efficient fast Primality testing algorithm to meet the requirement for generating the key in RSA algorithm. We use right-to-left-binary method for the exponent calculation. This reduces the number of cycles enhancing the performance of the system and reducing the area usage of the FPGA. These blocks are coded in Verilog and are synthesized and simulated in Xilinx 13.2 design suit.

Introducing Primality Testing Algorithm with an Implementation on 64 bits RSA Encryption Using Verilog

A new structure to develop 64-bit RSA encryption engine on FPGA is being presented in this paper that can be used as a standard device in the secured communication system. The RSA algorithm has three parts i.e. key generation, encryption and decryption. This procedure also requires random generation of prime numbers, therefore, we are proposing an efficient fast Primality testing algorithm to meet the requirement for generating the key in RSA algorithm. We use right-to-left-binary method for the exponent calculation. This reduces the number of cycles enhancing the performance of the system and reducing the area usage of the FPGA. These blocks are coded in Verilog and are synthesized and simulated in Xilinx 13.2 design suit.

3 Introducing Primality Testing Algorithm with an Implementation on 64 bits RSA Encryption Using Verilog

A new structure to develop 64-bit RSA encryption engine on FPGA is being presented in this paper that can be used as a standard device in the secured communication system. The RSA algorithm has three parts i.e. key generation, encryption and decryption. This procedure also requires random generation of prime numbers, therefore, we are proposing an efficient fast Primality testing algorithm to meet the requirement for generating the key in RSA algorithm. We use right-to-left-binary method for the exponent calculation. This reduces the number of cycles enhancing the performance of the system and reducing the area usage of the FPGA. These blocks are coded in Verilog and are synthesized and simulated in Xilinx 13.2 design suit.

A quantum primality test with order finding

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer N, the test tries to find an element of the multiplicative group of integers modulo N with order N-1. If one is found, the number is known to be prime. During the test, we can also show most of the times N is composite with certainty (and a witness) or, after \log\log N unsuccessful attempts to find an element of order N-1, declare it composite with high probability. The algorithm requires O((\log n)^2 n^3) operations for a number N with n bits, which can be reduced to O(\log\log n (\log n)^3 n^2) operations in the asymptotic limit if we use fast multiplication.

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/ $$\sqrt n $$ .

Improved error bounds for the Fermat primality test on random inputs

We investigate the probability that a random odd composite number passes a random Fermat primality test, improving on earlier estimates in moderate ranges. For example, with random numbers to $2^{200}$, our results improve on prior estimates by close to 3 orders of magnitude.