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Pratt Certificate
المؤلف:
Wagon, S
المصدر:
Mathematica in Action. New York: W. H. Freeman
الجزء والصفحة:
...
15-9-2020
1912
Pratt Certificate
The Pratt certificate is a primality certificate based on Fermat's little theorem converse. Prior to the work of Pratt (1975), the Lucas-Lehmer test had been regarded purely as a heuristic that worked a lot of the time (Knuth 1969). Pratt (1975) showed that Lehmer's primality heuristic could be made a nondeterministic procedure by applying it recursively to the factors of . As a consequence of this result, Pratt (1975) became the first to demonstrate that the resulting tree implies that prime factorization lies in the complexity class NP.
To generate a Pratt certificate, assume that is a positive integer and
{p_i}" src="https://mathworld.wolfram.com/images/equations/PrattCertificate/Inline3.gif" style="height:15px; width:23px" /> is the set of prime factors of
. Suppose there exists an integer
(called a "witness") such that
but
(mod
) whenever
is one of
. Then Fermat's little theorem converse states that
is prime (Wagon 1991, pp. 278-279).
By applying Fermat's little theorem converse to and recursively to each purported factor of
, a certificate for a given prime number can be generated. Stated another way, the Pratt certificate gives a proof that a number
is a primitive root of the multiplicative group (mod
) which, along with the fact that
has order
, proves that
is a prime.
The figure above gives a certificate for the primality of . The numbers to the right of the dashes are witnesses to the numbers to left. The set
{p_i}" src="https://mathworld.wolfram.com/images/equations/PrattCertificate/Inline20.gif" style="height:15px; width:23px" /> for
is given by
{2,37,107}" src="https://mathworld.wolfram.com/images/equations/PrattCertificate/Inline22.gif" style="height:15px; width:68px" />. Since
but
,
,
(mod 7919), 7 is a witness for 7919. The prime divisors of
are 2, 37, and 107. 2 is a so-called "self-witness" (i.e., it is recognized as a prime without further ado), and the remainder of the witnesses are shown as a nested tree. Together, they certify that 7919 is indeed prime. Because it requires the factorization of
, the method of Pratt certificates is best applied to small numbers (or those numbers
known to have easily factorable
).
A Pratt certificate is quicker to generate for small numbers than are other types of primality certificates. The Wolfram Language function ProvablePrimeQ[n] in the Wolfram Language package PrimalityProving` therefore generates an Atkin-Goldwasser-Kilian-Morain certificate only for numbers above a certain limit ( by default), and a Pratt certificate for smaller numbers.
REFERENCES:
Knuth, D. E. §4.5.4 in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. Reading, MA: Addison-Wesley, 1969.
Pratt, V. "Every Prime Has a Succinct Certificate." SIAM J. Comput. 4, 214-220, 1975.
Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 278-285, 1991.
Wilf, H. §4.10 in Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1986.
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