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Improved error bounds for the Fermat primality test on random inputs

Jared Duker Lichtman, Carl Pomerance
2018 Mathematics of Computation  
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.  ...  The first-named author is grateful for support received from the Byrne Scholars Program and the James O. Freedman Presidential Scholars program at Dartmouth College.  ... 
doi:10.1090/mcom/3314 fatcat:jqfawz5d4fdpvbnsl2e7pumxna

A Flexible and Efficient Algorithm for Generating Prime Numbers using Biometric Identity (Bio-PNGA)

B. Indrani, M. Karthigai
2016 International Journal of Computer Applications  
Most of the researchers have been made strong mathematical studies on primality testing and an observed progressive increase of cryptographic usages, prime number generation algorithms.  ...  A biometric based security system provides best on both authentication and confidentiality for public shared secret information.  ...  After one or more iterations, if n is not found to be a composite number, then it can be declared probably prime. 4) Fermat primality test The simplest probabilistic primality test is the Fermat primality  ... 
doi:10.5120/ijca2016912384 fatcat:4mpdd3krdvc23othbpjy2qpxiy

The twenty-fourth Fermat number is composite

Richard E. Crandall, Ernst W. Mayer, Jason S. Papadopoulos
2002 Mathematics of Computation  
The rigorous Pépin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors.  ...  For the sake of completeness, we also generated a Pépin residue for F 23 , and via the Suyama test determined that the known cofactor of this number is composite.  ...  just an Euler pseudoprime test-Pépin's contribution was to show that such a test in fact constitutes a rigorous primality test for the F n .  ... 
doi:10.1090/s0025-5718-02-01479-5 fatcat:oejuaintdfgopkevwmqdy76fbm

Generating Provable Primes Efficiently on Embedded Devices [chapter]

Christophe Clavier, Benoit Feix, Loïc Thierry, Pascal Paillier
2012 Lecture Notes in Computer Science  
Both our theoretical and experimental results show that constructive methods can generate provable primes essentially as efficiently as state-of-the-art generators for probable primes based on Fermat and  ...  This paper intends to provide practitioners with the first practical solutions for fast and secure generation of provable primes in embedded security devices.  ...  The authors would like to thank Vincent Verneuil for his valuable comments on this manuscript.  ... 
doi:10.1007/978-3-642-30057-8_22 fatcat:2dajdmtbz5c27dzdr3khkweroe

Almost all primes can be quickly certified

S Goldwasser, J Kilian
1986 Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86  
Thc test terminates in expected polynomial time on all but at most an exponentially vanishing fraction of the inputs of length k, for every k.  ...  The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on  ...  The primality testing algorithms used can be made to have such a small probability of error that, even with an exponential time penalty for making an error, the affect on the expected running time will  ... 
doi:10.1145/12130.12162 dblp:conf/stoc/GoldwasserK86 fatcat:r33yd4ohibd7vivthksverieya

Prime and Prejudice: Primality Testing Under Adversarial Conditions [article]

Martin R. Albrecht, Jake Massimo, Kenneth G. Paterson, Juraj Somorovsky
2018 IACR Cryptology ePrint Archive  
These give the number of iterations of Miller-Rabin needed for an error rate less than 2 −80 , when testing a random input n.  ...  This work provides a systematic analysis of primality testing under adversarial conditions, where the numbers being tested for primality are not generated randomly, but instead provided by a possibly malicious  ...  We thank Christian Elsholtz for initial guidance on the mathematical literature and Ian Miers for assistance with our analysis of OpenSSL.  ... 
dblp:journals/iacr/AlbrechtMPS18 fatcat:5fiyhuitczarlf2b5rhg3nvmie

Verification of the Miller–Rabin probabilistic primality test

Joe Hurd
2003 The Journal of Logic and Algebraic Programming  
Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the Miller-Rabin probabilistic primality test.  ...  Once verified, the primality test can either be executed in the logic (using rewriting) and used to prove the compositeness of numbers, or manually extracted to standard ML and used to find highly probable  ...  Finally, comments from the JLAP referees helped to improve many aspects of the paper.  ... 
doi:10.1016/s1567-8326(02)00065-6 fatcat:re6dekysq5cybj2doju4grkhnu

Optimized AKS Primality Testing: A Fluctuation Theory Perspective

Bhupendra Nath Tiwari, Jude Kibinde Kuipo, Joshua M. Adeegbe, Ninoslav Marina
2019 Cryptography  
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.  ...  parameters, we consider the randomized AKS primality testing function as the objective function.  ...  e della Ricerca, Italy for providing a stimulating environment for the realization of this research.  ... 
doi:10.3390/cryptography3020012 fatcat:eiowvss3mfdbzkyikackkhitam

Fast Effective Deterministic Primality Test Using CUDA/GPGPU

Abu Asaduzzaman, Anindya Maiti, Chok Meng Yip
There are many primality testing algorithms including mathematical models and computer programs. However, they are very time consuming when the given number n is very big or n→∞.  ...  whether an input number is prime or composite much faster.  ...  This suggests that the Fermat test for a prime is: pick a random a ∈ {1, ..., n−1 } and see if a n−1 =1 (mod n). If not, then n must be composite.  ... 
doi:10.24297/ijct.v12i3.3247 fatcat:wknbh2fk4rbthk5lrutq3oatra

Book Review: Inevitable randomness in discrete mathematics

J. Maurice Rojas
2013 Bulletin of the American Mathematical Society  
Acknowledgments I thank Peter Kuchment for valuable editorial remarks, and Leonid Gurvits, Dimitri Panchenko, and Joel Zinn for enlightening and enjoyable conversations.  ...  The Solovay-Strassen Primality Test, for an odd input N , then proceeds as follows: Pick a uniformly random a ∈ {1, . . . , N}.  ...  The preceding test gives one-sided error: a declaration of compositeness is always correct, but probable primality need not mean actual primality.  ... 
doi:10.1090/s0273-0979-2013-01407-1 fatcat:tnpuo33uqzdohmets6hu2pkfa4

Prime Proof Protocol [article]

Anna M. Johnston, Rathna Ramesh
2020 IACR Cryptology ePrint Archive  
Provable prime generation guarantees primality and is more efficient than probabilistic generation, and provides components for an efficient primality proof.  ...  Prime integers form the basis for finite field and elliptic curve cryptography, as well as many other applications.  ...  Let P be the integer we want to test for primality.  ... 
dblp:journals/iacr/JohnstonR20 fatcat:e7zp24cvyrhc5kgfwakwrurzdy

Designer Primes [article]

Anna M. Johnston
2020 IACR Cryptology ePrint Archive  
Unfortunately many systems use fixed primes for a variety of reasons, including the difficulty of generating trusted, random, cryptographically secure primes.  ...  This paper describes a variant of provable prime generation, intended for discrete logarithm based cryptography, based off Pocklington's theorem with improved efficiency, flexibility and security.  ...  value, h (j) ̸ = 0 for 0 ≤ j ≤ s Output: Modifies h ′ , Primality test Pocklington's theorem (theorem A.2) and the resulting primality tests require two conditions (equation 9, 8) and bounds on h (theorem  ... 
dblp:journals/iacr/Johnston20 fatcat:62nyeoi3pzhenftpyabpxnfflm

An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates

Ivan B. Damgård, Gudmund Skovbjerg Frandsen
2001 BRICS Report Series  
EQFT is well-suited for generating large, random prime numbers since on a random input number, it takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability.  ...  The variant has slightly larger worst case error probability than EQFT, but still improves on previous proposed tests.  ...  This is based on the fact that if the Fermat test was passed, we have a random z for which z n−1 = 1.  ... 
doi:10.7146/brics.v8i45.21705 fatcat:a4aug7zhnjbrjndbpkan3u2fvy

On a Modification of the Agrawal-Biswas Primality Test [article]

Hyun Jong Kim
2018 arXiv   pre-print
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.  ...  and T M R (N) respectively be the (upper bounds for the) runtimes of Algorithm 3 and the Miller-Rabin primality test.  ... 
arXiv:1810.09651v1 fatcat:lmretc7szrbttjeectafteiw6e

Page 2585 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews  
Some probabilistic tests for primality. Math. Sci. 14 (1989), no. 1, 55-61. This is a review of some efficient randomized algorithms for pri- mality testing.  ...  It is known that the Fermat congruence is a fairly good test for primeness. Choose a random odd integer n < x, choose a random integer b in 1<b<n-—1, and accept n if b"-' =1 mod n.  ... 
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