Bases for deterministic Miller-Rabin primality test
0
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Miller-Rabin primality test can be made deterministic when the number $n$ is small, for example "if $n < 2047$, it is enough to test [with base] $a = 2$".
How are those bases found? By brute force? Say I pick an upper bound 2047 and test many different bases to see if any of them always returns correct primality outcomes?
prime-numbers primality-test
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
$endgroup$
add a comment |
0
$begingroup$
Miller-Rabin primality test can be made deterministic when the number $n$ is small, for example "if $n < 2047$, it is enough to test [with base] $a = 2$".
How are those bases found? By brute force? Say I pick an upper bound 2047 and test many different bases to see if any of them always returns correct primality outcomes?
prime-numbers primality-test
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
$endgroup$
1
$begingroup$
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
$endgroup$
– gammatester
Nov 30 '18 at 19:17
1
$begingroup$
The Wikipedia references are much more up to date than the primes.utm page, but certainly the basic references such as Jaeschke are worthwhile as they go into detail about the mathematics (not relying on exhaustive testing). For the efficient deterministic bases, somewhere past 32-bit they rely on the Feitsma/Galway enumeration of all base-2 strong pseudoprimes to get deterministic results for 64-bit inputs, which makes searches practical.
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– DanaJ
Nov 30 '18 at 19:50
add a comment |
0
0
0
$begingroup$
Miller-Rabin primality test can be made deterministic when the number $n$ is small, for example "if $n < 2047$, it is enough to test [with base] $a = 2$".
How are those bases found? By brute force? Say I pick an upper bound 2047 and test many different bases to see if any of them always returns correct primality outcomes?
prime-numbers primality-test
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
$endgroup$
Miller-Rabin primality test can be made deterministic when the number $n$ is small, for example "if $n < 2047$, it is enough to test [with base] $a = 2$".
How are those bases found? By brute force? Say I pick an upper bound 2047 and test many different bases to see if any of them always returns correct primality outcomes?
prime-numbers primality-test
prime-numbers primality-test
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
408314
408314
1
$begingroup$
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
$endgroup$
– gammatester
Nov 30 '18 at 19:17
1
$begingroup$
The Wikipedia references are much more up to date than the primes.utm page, but certainly the basic references such as Jaeschke are worthwhile as they go into detail about the mathematics (not relying on exhaustive testing). For the efficient deterministic bases, somewhere past 32-bit they rely on the Feitsma/Galway enumeration of all base-2 strong pseudoprimes to get deterministic results for 64-bit inputs, which makes searches practical.
$endgroup$
– DanaJ
Nov 30 '18 at 19:50
add a comment |
1
$begingroup$
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
$endgroup$
– gammatester
Nov 30 '18 at 19:17
1
$begingroup$
The Wikipedia references are much more up to date than the primes.utm page, but certainly the basic references such as Jaeschke are worthwhile as they go into detail about the mathematics (not relying on exhaustive testing). For the efficient deterministic bases, somewhere past 32-bit they rely on the Feitsma/Galway enumeration of all base-2 strong pseudoprimes to get deterministic results for 64-bit inputs, which makes searches practical.
$endgroup$
– DanaJ
Nov 30 '18 at 19:50
1
1
$begingroup$
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
$endgroup$
– gammatester
Nov 30 '18 at 19:17
$begingroup$
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
$endgroup$
– gammatester
Nov 30 '18 at 19:17
1
1
$begingroup$
The Wikipedia references are much more up to date than the primes.utm page, but certainly the basic references such as Jaeschke are worthwhile as they go into detail about the mathematics (not relying on exhaustive testing). For the efficient deterministic bases, somewhere past 32-bit they rely on the Feitsma/Galway enumeration of all base-2 strong pseudoprimes to get deterministic results for 64-bit inputs, which makes searches practical.
$endgroup$
– DanaJ
Nov 30 '18 at 19:50
$begingroup$
The Wikipedia references are much more up to date than the primes.utm page, but certainly the basic references such as Jaeschke are worthwhile as they go into detail about the mathematics (not relying on exhaustive testing). For the efficient deterministic bases, somewhere past 32-bit they rely on the Feitsma/Galway enumeration of all base-2 strong pseudoprimes to get deterministic results for 64-bit inputs, which makes searches practical.
$endgroup$
– DanaJ
Nov 30 '18 at 19:50
add a comment |
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1
$begingroup$
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
$endgroup$
– gammatester
Nov 30 '18 at 19:17
1
$begingroup$
The Wikipedia references are much more up to date than the primes.utm page, but certainly the basic references such as Jaeschke are worthwhile as they go into detail about the mathematics (not relying on exhaustive testing). For the efficient deterministic bases, somewhere past 32-bit they rely on the Feitsma/Galway enumeration of all base-2 strong pseudoprimes to get deterministic results for 64-bit inputs, which makes searches practical.
$endgroup$
– DanaJ
Nov 30 '18 at 19:50