Yingpu Deng, Chang Lv
We propose an algorithm determining the primality of numbers $M=Ap^n+w_n$ where $w_n^{p-1}\equiv1\pmod{p^n}$ and $A<p^n$ and give example when $p=7$. $p$ th reciprocity law is involved. The algorithm runs in polynomial time in $\log_2(M)$ for fixed $p$ and $A$.
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Robert Denomme and Gordan Savin made a primality test for Fermat numbers 2^(2^k)+1 using elliptic curves. We propose another primality test using elliptic curves for Fermat numbers and also give primality tests for integers of the form 2^(2k+1)\pm2^(k+1)+1.
We propose some primality tests for 2^kn-1, where k, n in Z, k>= 2 and n odd. There are several tests depending on how big n is. These tests are proved using properties of elliptic curves. Essentially, the new primality tests are the elliptic curve version of the Lucas-Lehmer-Riesel primality test. Note:An anonymous referee suggested that Benedict H. Gross already proved the same result about a primality test for Mersenne primes using elliptic curve.
We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.
Gauss-Euler Primality Test
本文提出两种高效素性检验方法，可快速准确测试所有不超过 $2^{64}$ 的整数。
For an elliptic curve with CM by $K$ defined over its Hilbert class field, $E/H$, we extend Lenstra's finite fields test to generators of norms of certain ideals in $\mathcal{O}_H$, yielding a sufficient $\widetilde{O}(\log^3 N)$ primality test and partially answering an open question of Lemmermeyer in the case of CM elliptic curves. Letting $ι,γ, b\in \mathcal{O}_K$, $(ι)$ prime, and $b$ a primitive $k$-th root of unity modulo $(ι)^n$ we specialize this test to rational integers of the form $N_{K/\mathbb{Q}}(γι^n+b)$ with the norm of $γ$ small, giving a Las Vegas test for primality with avera…
We describe a primality test for number $M=(2p)^{2^n}+1$ with odd prime $p$ and positive integer $n$. And we also give the special primality criteria for all odd primes $p$ not exceeding 19. All these primality tests run in polynomial time in log$_{2}(M)$. A certain special $2p$-th reciprocity law is used to deduce our result.
Polynomial time primality tests for specific classes of numbers of the form $k\cdot 2^m \pm 1$ are introduced.
We study new primality tests based on linear recurrent sequences of degree two exploiting a matricial approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce a new specific primality test, which we will call generalized Pell test. We perform some numerical computations on the new primality tests and, for the generalized Pell test, we do not any pseudoprime up to $10^{10}$.
We present a variant of the Agrawal-Biswas algorithm, a Monte Carlo algorithm which tests the primality of an integer $N$ by checking whether or not $(x+a)^N$ and $x^N + a$ are equivalent in a residue ring of $\mathbb{Z}/N\mathbb{Z}[x]$. The variant that we present is also a randomization of Lenstra jr. and Pomerance's improvement to the Agrawal-Kayal-Saxena deterministic primality test. We show that our variant of the Agrawal-Biswas algorithm can be used with the Miller-Rabin primality test to yield an algorithm which is slower than the Miller-Rabin test but relatively more accurate.
We propose a pseudo-primality test using cyclic extensions of $\mathbb Z/n \mathbb Z$. For every positive integer $k \leq \log n$, this test achieves the security of $k$ Miller-Rabin tests at the cost of $k^{1/2+o(1)}$ Miller-Rabin tests.
We recall Charles Babbage's 1819 criterion for primality, based on simultaneous congruences for binomial coefficients, and extend it to a least-prime-factor test. We also prove a partial converse of his non-primality test, based on a single congruence. Two problems are posed. Along the way we encounter Bachet, Bernoulli, Bezout, Euler, Fermat, Kummer, Lagrange, Lucas, Vandermonde, Waring, Wilson, Wolstenholme, and several contemporary mathematicians.
At present one can not find a single counterexample to even a simplest version of Frobenius primality test. The assessment of probability of the mistake, presented in [I.B. Damgard and G.S.Frandsen, Journal of Cryptology, 2006] is strongly overestimated. In the present paper, the properties of simple divisors of FPP-numbers are proved. The lower bound for FPP are given.
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.
In Wilson's Theorem the primality of a number hinges on a congruence. We present a similar test where the primality of a number m hinges, instead, on the indivisibility of 4(m-5)! by m. One implication of this theorem is a necessary and sufficient condition for two numbers to be twin primes, a result reminiscent of Clement's theorem but involving indivisibility. MSC: 11A41 and 11A51.
In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost…
Novel performant primality test on a Pell's cubic
本研究提出一种基于佩尔三次曲线的新型素性测试算法，通过相关整数序列证明其必要素性条件，运算量随输入整数比特长度线性增长，对小于2的32次方的整数具有确定性，为公钥密码学提供了新的素性测试方案。
In this paper we generalize the classical Proth's theorem for integers of the form $N=Kp^n+1$. For these families, we present a primality test whose computational complexity is $\widetilde{O}(\log^2(N))$ and, what is more important, that requires only one modular exponentiation similar to that of Fermat's test. Consequently, the presented test improves the most often used one, derived from Pocklington's theorem, which usually requires the computation of several modular exponentiations together with some GCD's.
The use of the invariant's properties in the primality test and prime search
本文探讨了不变量性质在素性测试与素数搜索中的应用，提出了基于该性质的新算法，分析了其优缺点，并指出其能提供复合数因式分解的额外信息。
The Frobenius primality test is based on the properties of the Frobenius automorphism of the quadratic extension of the residue field. Although it is probabilistic, we show that is "very rarely wrong". To date there are no counterexamples to this method and there are reasons to believe that they do not exist at all. In this paper, we suggest a version of the Frobenius test and prove that it does not fail for numbers less than $2^{64}$. We also show that a "Frobenius pseudoprime" will necessarily have a prime divisor greater than 3000.
A primality test for $Kp^\ell - 1$ numbers
本文提出一种基于二次域的代数框架，用于推广Miller-Rabin素性测试，针对形式为$N = K p^{\ell} - 1$的整数设计确定性素性测试，仅需一次模幂运算，计算复杂度为$\tilde{\mathcal{O}}(\log^2 N)$，并建立Korselt准则的类比。
We describe an explicit generalized Lucasian test to determine the primality of numbers $h\cdot2^n\pm1$ when $h\nequiv0\pmod{17}$. This test is by means of fixed seeds which depend only on $h$. In particular when $h=16^m-1$ with $m$ odd, our paper gives a primality test with some fixed seeds depending only on $h$. Comparing the results of W. Bosma(1993) and P. Berrizbeitia and T. G. Berry(2004), our result adds new values of $h$ along with this line. Octic and bioctic reciprocity are used to deduce our result.
We develop a simple $O((\log n)^2)$ test as an extension of Proth's test for the primality for $p2^n+1$, $p>2^n$. This allows for the determination of large, non-Sierpinski primes $p$ and the smallest $n$ such that $p2^n+1$ is prime. If $p$ is a non-Sierpinski prime, then for all $n$ where $p2^n+1$ passes the initial test, $p2^n+1$ is prime with $3$ as a primitive root or is primover and divides the base $3$ Fermat Number, $GF(3,n-1)$. We determine the form the factors of any composite overpseudoprime that passes the initial test take by determining the form that factors of $GF(3,n-1)$ take.
This paper provides a proof of a LLT-like test for Fermat numbers, based on the properties of Lucas Sequences and on the method of Lehmer.
In this paper, two approximation algorithms are given. Let N be an odd composite number. The algorithms give new directions regarding primality test of given N. The first algorithm is given using a new method called digital coding method. It is conjectured that the algorithm finds a divisor of N in at most O(ln^{4} N), where ln denotes the logarithm with respect to base 2. The algorithm can be applied to find the next largest Mersenne prime number. Some directions are given regarding this. The second algorithm finds a prime divisor of N using the concept of graph pairs and it is proved that th…
The Baillie-PSW primality test combines Fermat and Lucas probable prime tests. It reports that a number is either composite or probably prime. No odd composite integer has been reported to pass this combination of primality tests if the parameters are chosen in an appropriate way. Here, we describe a significant strengthening of this test that comes at almost no additional computational cost. This is achieved by including in the test what we call Lucas-V pseudoprimes, of which there are only five less than $10^{15}$.
The Lucas-Lehmer (LL) primality test for Mersenne numbers is the fastest known primality test. In 1969, Hans Riesel published a modification of LL to test numbers of the form $N = h \cdot 2^n - 1$, where $h < 2^n$ is an odd integer and $n \ge 2$ \cite{Riesel}. This test is now known as the Lucas-Lehmer-Riesel (LLR) primality test. In Algorithm \ref{PrimalityAlgorithm}, we present a modification of LLR which works for any odd integer $N$. A probabilistic version of our algorithm runs in expected time $\tilde{O}(\log^3 N)$, and a deterministic version in expected $\tilde{O}(\log^4 N)$. We conclu…
A primality test for $Kp^n+1$ numbers and a generalization of Safe Primes and Sophie Germain Primes
本文对形式为$Kp^n+1$的整数推广了普罗特定理，提出了一种仅需一次模幂运算的质数测试方法，无需计算任何GCD。同时提供了两种测试以提高验证$Kp^n+1$数质数的可能性。作为这些测试的推论，我们给出了三个整数家族$N$，其质数性只能通过证明$a^{N-1} \equiv 1 \pmod N$（费马测试）来确认。还推广了安全质数，定义为$a$-安全质数，因其$N-1$具有与安全质数相同的大型质因数。我们讨论了这些数的分布问题，并提出了关于相关数$a$-索菲·热尔曼质数分布的猜想。
We use the arithmetic of the Kummer surface associated to the Jacobian of a hyperelliptic curve to study the primality of integers of the form $4m^2 5^n-1$. We provide an algorithm capable of proving the primality or compositeness of most of the integers in these families and discuss in detail the necessary steps to implement this algorithm in a computer. Although an indetermination is possible, in which case another choice of initial parameters should be used, we prove that the probability of reaching this situation is exceedingly low and decreases exponentially with $n$.
In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's converse of Fermat's little theorem. This result was considerably improved by Henry Pocklington in 1914 when he showed that it's not necessary to know all the prime factors of $n-1$ in order to determine if $n$ is prime. In this paper we optimize Pocklington's primality test for integers of the form $ap^{k}+1$ where $p$ is prime, $a<p$, $k\ge 1$. An extensi…
Lots and Lots of Perrin-Type Primality Tests and Their Pseudo-Primes
本研究借助Maple通过实验数学与符号计算搜寻Perrin型和Lucas型素性测试并筛选有效方法，还针对部分测试显式构造出无穷伪素数族，突破了过往仅能证明存在性的局限。
Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testi…
Generalized Cullen Numbers are positive integers of the form $C_b(n):=nb^n+1$. In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this family: $n^{b^{n}}\equiv (-1)^{b}$ (mod $nb^n+1$). It is stronger and has less computational cost than Fermat's test (for bases $b$ and $n$) and than Miller-Rabin's test (for base $n$). Pseudoprimes for this new test seem to be very scarce, only 4 pseudoprimes have been found among the many millions of Genera…
We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical "N-1" or "N+1" primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has a…
Taxonomy and Practical Evaluation of Primality Testing Algorithms
本文综述并分类了经典和量子算法用于因数分解与素性测试，分析其优劣、时间复杂度及应用比较，揭示量子算法在因数分解上的优势，但素性测试方面尚未有明显突破。
On the Eight Levels theorem and applications towards Lucas-Lehmer primality test for Mersenne primes, I
本文证明了八级定理，并基于此提出了三种 Lucas-Lehmer Mersenne 素性检验新版本及一个 Mersenne 合数新判据，以提升算法效率并促进新 Mersenne 素数的发现。
In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever we have complete analytic proof(s); otherwise the results are introduced as conjectures. In Part/Article 1, we start with the Baseline Primality Conjecture~(PBPC) which enables deterministic primality detection with a low complexity = O((log N)^2) ; when an explicit value of a Quadratic Non Residue (QNR) modulo-N is available (which happen…
Ordinary binary multiplication of natural numbers can be generalized in a non-trivial way to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with respect to ternary multiplication -- is defined, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields $\mathbb{Q}(\sqrt{-n})$, $n>0$, with odd discriminant and whose ring of integers admits unique factorization. We also describe how to determine representations of numbers as ternary products and related algorithms f…
In this paper, we study the properties of Carmichael numbers, false positives to several primality tests. We provide a classification for Carmichael numbers with a proportion of Fermat witnesses of less than 50%, based on if the smallest prime factor is greater than a determined lower bound. In addition, we conduct a Monte Carlo simulation as part of a probabilistic algorithm to detect if a given composite number is Carmichael. We modify this highly accurate algorithm with a deterministic primality test to create a novel, more efficient algorithm that differentiates between Carmichael numbers …
Three Brillhart-Lehmer-Selfridge primality proofs for Wagstaff numbers
本文基于Brillhart-Lehmer-Selfridge (BLS) N-1准则，对Wagstaff数W_2617、W_10501和W_12391进行了严格验证的素性证明，不依赖ECPP，仅使用经典N-1方法，并利用Cunningham项目表和FactorDB进行因子分解。
A primorial prime is a prime number of the form $p_n\# \pm 1$ where $p_n\#$ denotes the product of all primes less than or equal to $p_{n}$, the $n$-th prime. We show that the probability along the lines of Mertens' Theorem that either $p_n\# -1$ or $p_n\# +1$ is prime is $O(n^{-1})$ and that the probability that both $p_n\# -1$ and $p_n\# +1$ are prime is $O(n^{-2})$, for $n>1$. The latter result provides evidence that there are in total three instances where both $p_n\# -1$ and $p_n\# +1$ are prime. We provide proof that numbers of the from $p_n\# \pm 1$ have the highest probability of being…
Bad witnesses for a composite number
本文研究了多轮Miller-Rabin测试与Galois测试乘积的伪素性测试中坏见证者集合的平均大小，给出了其平均规模的描述。
In this paper we obtained an original integer sequence based on the properties of the multinomial coefficient. We investigated a property of the sequence that shows connection with a primality testing. For any prime n the n-th term in the sequence is less by 1 than the number of partitions of n. We hypothesize the existence of an asymptotic algorithm of primality testing.
Average Case Error Estimates of the Strong Lucas Test
针对强卢卡斯素性检验尚无平均情况误差界的问题，本文建立了其平均情况误差界，给出随机奇k比特数经t轮检验返回合数的概率上界，提出排除小素数倍数的改进算法并分类高贡献数，提升检验应用信心。
From Worst to Average Case to Incremental Search Bounds of the Strong Lucas Test
本文研究强Lucas测试的错误界，证明通过t轮连续测试及额外标准测试的整数为素数的概率大于1-(4/15)^t，并首次给出基于强Lucas测试的增量搜索算法的错误界。
In this paper, we propose a new primality test, and then we employ this test to find a formula for π that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number
We give a new sufficient condition which allows to test primality of Fermat's numbers. This characterization uses uniquely values at most equal to tested Fermat number. The robustness of this result is due to a strict use of elementary arithmetic technical tools and it will be susceptible to open gates for revolutionary statement that all Fermat's numbers are all decomposable.
We give Deterministic Primality tests for large families of numbers. These tests were inspired in the recent and celebrated Agrawal-Kayal-Saxena (AKS) test. The AKS test has proved polynomial complexity O ((log n)^12) and they expect it to be O ((log n)^6) . Our tests have proved complexity O ((log n)^6). The complexity decreases to O ((log n)^4) as the power of 2 dividing n + 1 or n - 1 increases. On large enough primes, our tests, in their worst case, run at least 2^9 times faster than the AKS test.
Definition of the invariant and the relationship with the compounds numbers. Generalisation of the Euler theorem
本文引入不变量的概念及其性质用于素性检验，结合欧拉定理将其推广至所有 $a^{φ(m)}$ 情形，并提出替代 Totient 函数以解释 Carmichael 数的新假设。
We define a variant of the Miller-Rabin primality test, which is in between Miller-Rabin and Fermat in terms of strength. We show that this test has infinitely many "Carmichael" numbers. We show that the test can also be thought of as a variant of the Solovay-Strassen test. We explore the growth of the test's "Carmichael" numbers, giving some empirical results and a discussion of one particularly strong pattern which appears in the results.
The structure of the group $(\mathbb{Z}/n\mathbb{Z})^\star$ and Fermat's little theorem are the basis for some of best-known primality testing algorithms. Many related concepts arise: Euler's totient function and Carmichael's lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer's totient problem, Giuga's conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group $\mathcal{G}_n:=\{a+bi\in\mathbb{Z}[i]/n\mathbb{Z}[i] : a^2+b^2\equiv 1\ \textrm{$\pmod n$}\}$. In particular we characterize Gau…
The strong probable primality test is an important practical tool for discovering prime numbers. Its effectiveness derives from the following fact: for any odd composite number $n$, if a base $a$ is chosen at random, the algorithm is unlikely to claim that $n$ is prime. If this does happen we call $a$ a liar. In 1986, Erdős and Pomerance computed the normal and average number of liars, over all $n \leq x$. We continue this theme and use a variety of techniques to count $n \leq x$ with exactly two strong liars, those being the $n$ for which the strong test is maximally effective. We evaluate th…
Motivated by Euler's observation that the polynomial $x^{2} + x + 41$ takes on prime values for $0 \leq x \leq 39$, we search for large values of $x$ for which $N = x^{2} + x + 41$ is prime. To apply classical primality proving results based on the factorization of $N-1$, we choose $x$ to have the form $g(y)$, chosen so that $g(y)^{2} + g(y) + 40$ is reducible. Our main result is an explicit, 60,000 digit prime number of the form $x^{2} + x + 41$.
We will describe an algorithm to construct an elliptic curve $E_{f_q}$ over some prime field $\mathbb{F}_p$ such that such that $|E_{f_q}(\mathbb{F}_p)| = f_q$, where $f_q$ is a probable Fibonacci prime for some prime index $q$. The algorithm is a variant of the efficient CM-construction by Br$\ddot{o}$ker and Stevenhagen, which is well suited for Fibonacci primes due to their arithmetic properties. The time complexity of our algorithm is expected to be lower than $\widetilde{O}(\log^3({f_q}))$. The construction process is a series of algorithms, where each is a test for primality.
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a probability as close to 1 as desired. The algorithm requires $O((\log n )^3n^3)$ quantum operations, or $O(\log\log n (\log n)^4 n^2)$ operations using fast multiplication. Verification, quantum optimizations and applications to RSA and primality tests are also discussed.
Ours go to 211: Euler pseudoprimes to 47 prime bases (from Carmichael numbers)
本文研究了卡迈克尔数中包含的欧拉伪素数，并提出了一种快速算法来生成新的欧拉伪素数，其中某些伪素数能够通过前47个素数基的测试。
A Deterministic Cryptographic Prime Generation Chain over Monogenic Cubic Number Fields and their Generalizations
提出一种从已知素数构造新素数的直接方法，利用单项纯三次域结构，仅需一次模幂运算即可在拟二次时间内完成素性证明，并推广到任意素数次数的纯数域。
In this paper, we look at numbers of the form $H_{r,k}:=F_{k-1}F_{r-k+2}+F_{k}F_{r-k}$. These numbers are the entries of a triangular array called the \emph{determinant Hosoya triangle} which we denote by ${\mathcal H}$. We discuss the divisibility properties of the above numbers and their primality. We give a small sieve of primes to illustrate the density of prime numbers in ${\mathcal H}$. Since the Fibonacci and Lucas numbers appear as entries in ${\mathcal H}$, our research is an extension of the classical questions concerning whether there are infinitely many Fibonacci or Lucas primes. W…
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 …
In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly random $a<q$ coprime to $q$, and choose $p$ of the form $a+t\cdot q$, where only $t$ is updated if the primality test fails. We prove that variants of this approach provide prime generation algorithms requiring few random bits and whose output distribution is close to uniform, under less and less expensive assumptions: first a relatively strong conjecture by H.…
Given a large positive integer $N$, how quickly can one construct a prime number larger than $N$ (or between $N$ and 2N)? Using probabilistic methods, one can obtain a prime number in time at most $\log^{O(1)} N$ with high probability by selecting numbers between $N$ and 2N at random and testing each one in turn for primality until a prime is discovered. However, if one seeks a deterministic method, then the problem is much more difficult, unless one assumes some unproven conjectures in number theory; brute force methods give a $O(N^{1+o(1)})$ algorithm, and the best unconditional algorithm, d…
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