Bases for deterministic Miller-Rabin primality test
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
406314
add a comment |
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
406314
1
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
– gammatester
Nov 30 '18 at 19:17
1
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.
– DanaJ
Nov 30 '18 at 19:50
add a comment |
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
406314
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
406314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
406314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
406314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
406314
asked Nov 30 '18 at 18:45
Ecir HanaEcir Hana
406314
406314
1
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
– gammatester
Nov 30 '18 at 19:17
1
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.
– DanaJ
Nov 30 '18 at 19:50
add a comment |
1
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
– gammatester
Nov 30 '18 at 19:17
1
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.
– DanaJ
Nov 30 '18 at 19:50
1
1
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
– gammatester
Nov 30 '18 at 19:17
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
– gammatester
Nov 30 '18 at 19:17
1
1
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.
– DanaJ
Nov 30 '18 at 19:50
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.
– DanaJ
Nov 30 '18 at 19:50
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3020478%2fbases-for-deterministic-miller-rabin-primality-test%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3020478%2fbases-for-deterministic-miller-rabin-primality-test%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
May be Finding primes & proving primality and the listed references help; especially Jaeschke's paper On strong pseudoprimes to several bases
– gammatester
Nov 30 '18 at 19:17
1
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.
– DanaJ
Nov 30 '18 at 19:50