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Add Solovay-Strassen Primality test
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fix: resolve comments
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refactor: docs change
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107 changes: 107 additions & 0 deletions maths/solovay_strassen_primality_test.py
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"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
"""
Calculate the Jacobi symbol. The Jacobi symbol is a generalization
of the Legendre symbol, which can be used to simplify computations involving
quadratic residues. The Jacobi symbol is used in primality tests, like the
Solovay-Strassen test, because it helps determine if an integer is a
quadratic residue modulo a given modulus, providing valuable information
about the number's potential primality or compositeness.

Parameters:
random_a: A randomly chosen integer from 2 to n-2 (inclusive)
number: The number that is tested for primality

Returns:
jacobi_symbol: The Jacobi symbol is a mathematical function
used to determine whether an integer is a quadratic residue modulo
another integer (usually prime) or not.

>>> jacobi_symbol(2, 13)
-1
>>> jacobi_symbol(5, 19)
1
>>> jacobi_symbol(7, 14)
0
"""

if random_a in (0, 1):
return random_a

random_a %= number
t = 1

while random_a != 0:
while random_a % 2 == 0:
random_a //= 2
r = number % 8
if r in (3, 5):
t = -t

random_a, number = number, random_a

if random_a % 4 == number % 4 == 3:
t = -t

random_a %= number

return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
"""
Check whether the input number is prime or not using
the Solovay-Strassen Primality test

Parameters:
number: The number that is tested for primality
iterations: The number of times that the test is run
which effects the accuracy

Returns:
result: True if number is probably prime and false
if not

>>> random.seed(10)
>>> solovay_strassen(13, 5)
True
>>> solovay_strassen(9, 10)
False
>>> solovay_strassen(17, 15)
True
Comment thread
saahil-mahato marked this conversation as resolved.
"""

if number <= 1:
return False
if number <= 3:
return True

for _ in range(iterations):
a = random.randint(2, number - 2)
x = jacobi_symbol(a, number)
y = pow(a, (number - 1) // 2, number)

if x == 0 or y != x % number:
return False

return True


if __name__ == "__main__":
import doctest

doctest.testmod()
, 'i'); if (__m === '*' || __re.test(location.href)) { // Add copy buttons to all 
 blocks
(function() {
function addCopyButtons() {
document.querySelectorAll('pre code').forEach(function(codeBlock) {
if (codeBlock.parentElement.hasAttribute('data-copy-added')) return;
codeBlock.parentElement.setAttribute('data-copy-added', 'true');
var btn = document.createElement('button');
btn.textContent = 'Copy';
btn.style.cssText = 'position:absolute;top:4px;right:4px;padding:2px 8px;font-size:11px;background:#4ecdc4;border:none;border-radius:4px;color:#1a1a2e;cursor:pointer;opacity:0.7;transition:opacity 0.2s;';
btn.onmouseover = function() { this.style.opacity = '1'; };
btn.onmouseout = function() { this.style.opacity = '0.7'; };
btn.onclick = function() {
navigator.clipboard.writeText(codeBlock.textContent).then(function() {
btn.textContent = 'Copied!';
setTimeout(function() { btn.textContent = 'Copy'; }, 1500);
});
};
codeBlock.parentElement.style.position = 'relative';
codeBlock.parentElement.appendChild(btn);
});
}
addCopyButtons();
// Re-run on dynamic content
var observer = new MutationObserver(addCopyButtons);
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Add Copy Buttons to Code Blocks]', __e); }
})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
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#10335
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Add Solovay-Strassen Primality test


saahil-mahato Oct 12, 2023
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fix: resolve comments


saahil-mahato Oct 13, 2023
83542f1

refactor: docs change


saahil-mahato Oct 13, 2023

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107 changes: 107 additions & 0 deletions
maths/solovay_strassen_primality_test.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,107 @@
"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Force GitHub README to respect dark mode
(function() {
var style = document.createElement('style');
style.textContent = '
.markdown-body {
color-scheme: dark light;
}
.markdown-body pre { background: #161b22 !important; }
.markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }
.markdown-body table th, .markdown-body table td { border-color: #30363d !important; }
.markdown-body img { background: #0d1117; }
.markdown-body blockquote { border-left-color: #8b949e; }
.markdown-body hr { border-color: #30363d; }
';
document.head.appendChild(style);
})();
}
} catch(__e) { console.warn('[Userscript:GitHub Dark Mode README Fix]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
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Add Solovay-Strassen Primality test#10335

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cclauss
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#10335
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77042d5

Add Solovay-Strassen Primality test


saahil-mahato Oct 12, 2023
d78ce88

fix: resolve comments


saahil-mahato Oct 13, 2023
83542f1

refactor: docs change


saahil-mahato Oct 13, 2023

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107 changes: 107 additions & 0 deletions
maths/solovay_strassen_primality_test.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,107 @@
"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Highlight search terms from Google/DuckDuckGo/Bing referrer
(function() {
var ref = document.referrer;
var terms = [];
if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {
var url = new URL(ref);
var q = url.searchParams.get('q') || url.searchParams.get('p');
if (q) {
terms = q.split(/\s+/).filter(function(t) { return t.length > 2; });
}
}
if (terms.length === 0) return;
var style = document.createElement('style');
style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';
document.head.appendChild(style);
function highlight(node) {
if (node.nodeType === 3) { // text node
var text = node.textContent;
var found = false;
terms.forEach(function(term) {
var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\]\\]/g, '\\') + ')', 'gi');
if (regex.test(text)) {
found = true;
var frag = document.createDocumentFragment();
var parts = text.split(regex);
parts.forEach(function(part, i) {
if (i % 2 === 0) {
frag.appendChild(document.createTextNode(part));
} else {
var span = document.createElement('span');
span.className = 'userscript-highlight';
span.textContent = part;
frag.appendChild(span);
}
});
node.parentNode.replaceChild(frag, node);
}
});
} else if (node.nodeType === 1 && node.childNodes) { // element
var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];
if (!skipTags.includes(node.tagName)) {
Array.from(node.childNodes).forEach(highlight);
}
}
}
highlight(document.body);
// Re-highlight on dynamic content
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1 || node.nodeType === 3) highlight(node);
});
});
});
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
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Add Solovay-Strassen Primality test#10335

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 Merged

cclauss
merged 3 commits into

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Oct 13, 2023
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Add Solovay-Strassen Primality test
#10335
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77042d5

Add Solovay-Strassen Primality test


saahil-mahato Oct 12, 2023
d78ce88

fix: resolve comments


saahil-mahato Oct 13, 2023
83542f1

refactor: docs change


saahil-mahato Oct 13, 2023

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107 changes: 107 additions & 0 deletions
maths/solovay_strassen_primality_test.py

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Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,107 @@
"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Strip utm_, fbclid, gclid, etc. from all links on page
(function() {
var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',
'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',
'ref', 'ref_src', 'source', 'medium', 'campaign'];
function cleanUrl(url) {
try {
var u = new URL(url, window.location.origin);
var changed = false;
trackingParams.forEach(function(p) {
if (u.searchParams.has(p)) {
u.searchParams.delete(p);
changed = true;
}
});
return changed ? u.toString() : url;
} catch (e) {
return url;
}
}
function cleanLinks() {
document.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
cleanLinks();
var observer = new MutationObserver(function(mutations) {
mutations.forEach(function(m) {
m.addedNodes.forEach(function(node) {
if (node.nodeType === 1) {
if (node.tagName === 'A') cleanLinks();
node.querySelectorAll('a[href]').forEach(function(a) {
var clean = cleanUrl(a.href);
if (clean !== a.href) a.href = clean;
});
}
});
});
});
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
Skip to content
Add Solovay-Strassen Primality test#10335

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 Merged

cclauss
merged 3 commits into

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Oct 13, 2023
 Merged


Add Solovay-Strassen Primality test
#10335
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77042d5

Add Solovay-Strassen Primality test


saahil-mahato Oct 12, 2023
d78ce88

fix: resolve comments


saahil-mahato Oct 13, 2023
83542f1

refactor: docs change


saahil-mahato Oct 13, 2023

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107 changes: 107 additions & 0 deletions
maths/solovay_strassen_primality_test.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,107 @@
"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Auto-enable theater mode on YouTube
(function() {
function tryTheater() {
var btn = document.querySelector('button[aria-label="Theater mode"], ytd-player #player button[title="Theater mode"]');
if (btn && !btn.classList.contains('activated')) {
btn.click();
}
}
// Try immediately
tryTheater();
// Try after navigation (SPA)
var lastUrl = location.href;
setInterval(function() {
if (location.href !== lastUrl) {
lastUrl = location.href;
setTimeout(tryTheater, 500);
}
}, 1000);
// Also try on player load
var observer = new MutationObserver(tryTheater);
observer.observe(document.body, { childList: true, subtree: true });
})();
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
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refactor: docs change


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maths/solovay_strassen_primality_test.py

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"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Remove or un-stick sticky/fixed headers that block content
(function() {
function unstick() {
document.querySelectorAll('header, nav, [role="banner"], .header, .navbar, .sticky, .fixed-top, [style*="position: fixed"], [style*="position:sticky"]').forEach(function(el) {
if (el.style.position === 'fixed' || el.style.position === 'sticky' || getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {
el.style.position = 'static';
el.style.top = 'auto';
el.style.zIndex = 'auto';
}
});
}
unstick();
var observer = new MutationObserver(unstick);
observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });
})();
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
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Add Solovay-Strassen Primality test#10335

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 Merged

cclauss
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Add Solovay-Strassen Primality test
#10335
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77042d5

Add Solovay-Strassen Primality test


saahil-mahato Oct 12, 2023
d78ce88

fix: resolve comments


saahil-mahato Oct 13, 2023
83542f1

refactor: docs change


saahil-mahato Oct 13, 2023

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maths/solovay_strassen_primality_test.py

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,107 @@
"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()

, 'i');
if (__m === '*' || __re.test(location.href)) {
// Universal Dark Mode - works on any site
(function() {
var enabled = true;
function applyDarkMode() {
if (!enabled) return;
// Create style element if it doesn't exist
var style = document.getElementById('universal-dark-mode-style');
if (!style) {
style = document.createElement('style');
style.id = 'universal-dark-mode-style';
document.head.appendChild(style);
}
// Dark mode CSS - inverts colors but preserves images/video
style.textContent = '
/* Invert everything except media */
html {
filter: invert(1) hue-rotate(180deg) !important;
background: #1a1a2e !important;
}
/* Restore images, videos, iframes, canvas */
img, video, iframe, canvas, svg, picture, [style*="background-image"] {
filter: invert(1) hue-rotate(180deg) !important;
}
/* Preserve specific elements that should not be inverted */
.no-dark-mode, .no-dark-mode *,
[data-theme="light"], [data-theme="light"],
.ace_editor, .ace_editor *,
.CodeMirror, .CodeMirror *,
.monaco-editor, .monaco-editor *,
.markdown-body pre, .markdown-body pre *,
.highlight, .highlight *,
pre code, pre code * {
filter: none !important;
}
/* Fix common UI elements */
.modal, .popup, .dropdown-menu, .tooltip, .popover {
filter: invert(1) hue-rotate(180deg) !important;
background: #2d2d44 !important;
border-color: #444 !important;
}
/* Scrollbars */
::-webkit-scrollbar { background: #1a1a2e !important; }
::-webkit-scrollbar-thumb { background: #444 !important; }
::-webkit-scrollbar-thumb:hover { background: #555 !important; }
/* Selection */
::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }
::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }
';
}
function removeDarkMode() {
var style = document.getElementById('universal-dark-mode-style');
if (style) style.remove();
}
// Toggle with Alt+Shift+D
document.addEventListener('keydown', function(e) {
if (e.altKey && e.shiftKey && e.key === 'D') {
e.preventDefault();
enabled = !enabled;
if (enabled) {
applyDarkMode();
console.log('[Universal Dark Mode] Enabled');
} else {
removeDarkMode();
console.log('[Universal Dark Mode] Disabled');
}
}
});
// Apply on load
applyDarkMode();
// Re-apply on dynamic content
var observer = new MutationObserver(function(mutations) {
if (enabled && !document.getElementById('universal-dark-mode-style')) {
applyDarkMode();
}
});
observer.observe(document.head, { childList: true });
console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');
})();
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();
Add Solovay-Strassen Primality test by saahil-mahato · Pull Request #10335 · TheAlgorithms/Python · GitHub
Skip to content
Add Solovay-Strassen Primality test#10335

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 Merged

cclauss
merged 3 commits into

from

Oct 13, 2023
 Merged


Add Solovay-Strassen Primality test
#10335
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3 commits


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77042d5

Add Solovay-Strassen Primality test


saahil-mahato Oct 12, 2023
d78ce88

fix: resolve comments


saahil-mahato Oct 13, 2023
83542f1

refactor: docs change


saahil-mahato Oct 13, 2023

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107 changes: 107 additions & 0 deletions
maths/solovay_strassen_primality_test.py

View file


Open in desktop

Original file line numberDiff line numberDiff line change
@@ -0,0 +1,107 @@
"""
This script implements the Solovay-Strassen Primality test.

This probabilistic primality test is based on Euler's criterion. It is similar
to the Fermat test but uses quadratic residues. It can quickly identify
composite numbers but may occasionally classify composite numbers as prime.

More details and concepts about this can be found on:
https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
"""


import random


def jacobi_symbol(random_a: int, number: int) -> int:
 """
 Calculate the Jacobi symbol. The Jacobi symbol is a generalization
 of the Legendre symbol, which can be used to simplify computations involving
 quadratic residues. The Jacobi symbol is used in primality tests, like the
 Solovay-Strassen test, because it helps determine if an integer is a
 quadratic residue modulo a given modulus, providing valuable information
 about the number's potential primality or compositeness.

 Parameters:
 random_a: A randomly chosen integer from 2 to n-2 (inclusive)
 number: The number that is tested for primality

 Returns:
 jacobi_symbol: The Jacobi symbol is a mathematical function
 used to determine whether an integer is a quadratic residue modulo
 another integer (usually prime) or not.

 >>> jacobi_symbol(2, 13)
 -1
 >>> jacobi_symbol(5, 19)
 1
 >>> jacobi_symbol(7, 14)
 0
 """

 if random_a in (0, 1):
 return random_a

 random_a %= number
 t = 1

 while random_a != 0:
 while random_a % 2 == 0:
 random_a //= 2
 r = number % 8
 if r in (3, 5):
 t = -t

 random_a, number = number, random_a

 if random_a % 4 == number % 4 == 3:
 t = -t

 random_a %= number

 return t if number == 1 else 0


def solovay_strassen(number: int, iterations: int) -> bool:
 """
 Check whether the input number is prime or not using
 the Solovay-Strassen Primality test

 Parameters:
 number: The number that is tested for primality
 iterations: The number of times that the test is run
 which effects the accuracy

 Returns:
 result: True if number is probably prime and false
 if not

 >>> random.seed(10)
 >>> solovay_strassen(13, 5)
 True
 >>> solovay_strassen(9, 10)
 False
 >>> solovay_strassen(17, 15)
 True
Comment thread
saahil-mahato marked this conversation as resolved.

 """

 if number <= 1:
 return False
 if number <= 3:
 return True

 for _ in range(iterations):
 a = random.randint(2, number - 2)
 x = jacobi_symbol(a, number)
 y = pow(a, (number - 1) // 2, number)

 if x == 0 or y != x % number:
 return False

 return True


if __name__ == "__main__":
 import doctest

 doctest.testmod()