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Improved error bounds for the Fermat primality test on random inputs

2018
*
Mathematics of Computation
*

We investigate

doi:10.1090/mcom/3314
fatcat:jqfawz5d4fdpvbnsl2e7pumxna
*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. ...##
###
A Flexible and Efficient Algorithm for Generating Prime Numbers using Biometric Identity (Bio-PNGA)

2016
*
International Journal of Computer Applications
*

Most of

doi:10.5120/ijca2016912384
fatcat:4mpdd3krdvc23othbpjy2qpxiy
*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*...##
###
The twenty-fourth Fermat number is composite

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 . ...

##
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Generating Provable Primes Efficiently on Embedded Devices
[chapter]

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-

doi:10.1007/978-3-642-30057-8_22
fatcat:2dajdmtbz5c27dzdr3khkweroe
*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. ...##
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Almost all primes can be quickly certified

1986
*
Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86
*

Thc

doi:10.1145/12130.12162
dblp:conf/stoc/GoldwasserK86
fatcat:r33yd4ohibd7vivthksverieya
*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 ...##
###
Prime and Prejudice: Primality Testing Under Adversarial Conditions
[article]

2018
*
IACR Cryptology ePrint Archive
*

These give

dblp:journals/iacr/AlbrechtMPS18
fatcat:5fiyhuitczarlf2b5rhg3nvmie
*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. ...##
###
Verification of the Miller–Rabin probabilistic primality test

2003
*
The Journal of Logic and Algebraic Programming
*

Using

doi:10.1016/s1567-8326(02)00065-6
fatcat:re6dekysq5cybj2doju4grkhnu
*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. ...##
###
Optimized AKS Primality Testing: A Fluctuation Theory Perspective

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. ...

##
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Fast Effective Deterministic Primality Test Using CUDA/GPGPU

2013
*
INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY
*

There are many

doi:10.24297/ijct.v12i3.3247
fatcat:wknbh2fk4rbthk5lrutq3oatra
*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. ...##
###
Book Review: Inevitable randomness in discrete mathematics

2013
*
Bulletin of the American Mathematical Society
*

Acknowledgments I thank Peter Kuchment

doi:10.1090/s0273-0979-2013-01407-1
fatcat:tnpuo33uqzdohmets6hu2pkfa4
*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*. ...##
###
Prime Proof Protocol
[article]

2020
*
IACR Cryptology ePrint Archive
*

Provable prime generation guarantees

dblp:journals/iacr/JohnstonR20
fatcat:e7zp24cvyrhc5kgfwakwrurzdy
*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*. ...##
###
Designer Primes
[article]

2020
*
IACR Cryptology ePrint Archive
*

Unfortunately many systems use fixed primes

dblp:journals/iacr/Johnston20
fatcat:62nyeoi3pzhenftpyabpxnfflm
*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 ...##
###
An Extended Quadratic Frobenius Primality Test with Average Case Error Estimates

2001
*
BRICS Report Series
*

EQFT is well-suited

doi:10.7146/brics.v8i45.21705
fatcat:a4aug7zhnjbrjndbpkan3u2fvy
*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. ...##
###
On a Modification of the Agrawal-Biswas Primality Test
[article]

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*. ...

##
###
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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