Algorithms: Number Theory

"use strict";
// Extended Euclid Modular Inverse in JavaScript
//
// PURPOSE
// Extended Euclid tracks a Bezout coefficient while computing the gcd. When gcd(a,m)=1, that
// coefficient is the modular inverse.
//
// METHOD
// Maintain consecutive remainders and coefficients. Update both with the Euclidean quotient;
// normalize the resulting coefficient into [0,m).
//
// COMPLEXITY
// O(log m) arithmetic steps and O(1) auxiliary space.
//
// CONTRACT
// 0 <= a < modulus, 2 <= modulus <= 10000. Returns -1 if no inverse exists. This bounded
// integer implementation is not constant time or suitable for cryptographic secrets.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// Node.js 18+: Int32Array inputs and workspaces provide compact integer storage. Automatic
// lengths and checked arithmetic reject values outside signed 32-bit. Invalid inputs or
// overflow throw RangeError.
//
// EXAMPLE
// Expected return: 4.
// Run: node example.js
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(valid) { if (!valid) throw new RangeError("invalid algorithm input"); }
function alg_checked(v) { if (!Number.isInteger(v) || v < -2147483648 || v > 2147483647) throw new RangeError("32-bit result overflow"); return v; }
function alg_add(a,b) { return alg_checked(a+b); }
function alg_sub(a,b) { return alg_checked(a-b); }
function alg_mul(a,b) { return alg_checked(a*b); }
function alg_size(n) { alg_require(Number.isInteger(n) && n>=0 && n<=4000000); return n; }

function modular_inverse(a, modulus) {
    alg_require(Number.isInteger(a) && a >= -2147483648 && a <= 2147483647);
    alg_require(Number.isInteger(modulus) && modulus >= -2147483648 && modulus <= 2147483647);
    alg_require(((modulus >= 2) && (modulus <= 10000) && (a >= 0) && (a < modulus)));
    let old_r = modulus;
    let r = a;
    let old_t = 0;
    let t = 1;
    let quotient = 0;
    let temp = 0;
    while ((r != 0)) {
        quotient = Math.floor(old_r / r);
        temp = alg_sub(old_r, alg_mul(quotient, r));
        old_r = r;
        r = temp;
        temp = alg_sub(old_t, alg_mul(quotient, t));
        old_t = t;
        t = temp;
    }
    if ((old_r != 1)) {
        return alg_sub(0, 1);
    }
    return (alg_add((old_t % modulus), modulus) % modulus);
}

let a = 3;
let modulus = 11;
let result = modular_inverse(a, modulus);
console.log(result);

from bisect import bisect_left

# Extended Euclid Modular Inverse in Python
#
# PURPOSE
# Extended Euclid tracks a Bezout coefficient while computing the gcd. When gcd(a,m)=1, that
# coefficient is the modular inverse.
#
# METHOD
# Maintain consecutive remainders and coefficients. Update both with the Euclidean quotient;
# normalize the resulting coefficient into [0,m).
#
# COMPLEXITY
# O(log m) arithmetic steps and O(1) auxiliary space.
#
# CONTRACT
# 0 <= a < modulus, 2 <= modulus <= 10000. Returns -1 if no inverse exists. This bounded
# integer implementation is not constant time or suitable for cryptographic secrets.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 4.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def modular_inverse(a: int, modulus: int) -> int:
    alg_require(type(a) is int and -2147483648 <= a <= 2147483647)
    alg_require(type(modulus) is int and -2147483648 <= modulus <= 2147483647)
    alg_require(modulus >= 2 and modulus <= 10000 and (a >= 0) and (a < modulus))
    old_r: int = modulus
    r: int = a
    old_t: int = 0
    t: int = 1
    quotient: int = 0
    temp: int = 0
    while r != 0:
        quotient = old_r // r
        temp = old_r - quotient * r
        old_r = r
        r = temp
        temp = old_t - quotient * t
        old_t = t
        t = temp
    if old_r != 1:
        return -1
    return (old_t % modulus + modulus) % modulus

a = 3
modulus = 11
result = modular_inverse(a, modulus)
print(result)

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Extended Euclid Modular Inverse in C
//
// PURPOSE
// Extended Euclid tracks a Bezout coefficient while computing the gcd. When gcd(a,m)=1, that
// coefficient is the modular inverse.
//
// METHOD
// Maintain consecutive remainders and coefficients. Update both with the Euclidean quotient;
// normalize the resulting coefficient into [0,m).
//
// COMPLEXITY
// O(log m) arithmetic steps and O(1) auxiliary space.
//
// CONTRACT
// 0 <= a < modulus, 2 <= modulus <= 10000. Returns -1 if no inverse exists. This bounded
// integer implementation is not constant time or suitable for cryptographic secrets.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 4.
// Run: gcc -std=c99 -O2 example.c -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int modular_inverse(int a, int modulus) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    if (!(((modulus >= 2) && (modulus <= 10000) && (a >= 0) && (a < modulus)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int old_r = modulus;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int r = a;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int old_t = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int t = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int quotient = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int temp = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (r != 0);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        quotient = (old_r / r);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        temp = alg_sub(old_r, alg_mul(quotient, r, &alg_error), &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        old_r = r;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        r = temp;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        temp = alg_sub(old_t, alg_mul(quotient, t, &alg_error), &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        old_t = t;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        t = temp;
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    int alg_condition_1 = (old_r != 1);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_condition_1) {
        alg_result = alg_sub(0, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        goto alg_cleanup;
    }
    alg_result = (alg_add((old_t % modulus), modulus, &alg_error) % modulus);
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    return alg_result;
}

int main(void) {
    int a = 3;
    int modulus = 11;
    int result = modular_inverse(a, modulus);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Extended Euclid Modular Inverse in C++
//
// PURPOSE
// Extended Euclid tracks a Bezout coefficient while computing the gcd. When gcd(a,m)=1, that
// coefficient is the modular inverse.
//
// METHOD
// Maintain consecutive remainders and coefficients. Update both with the Euclidean quotient;
// normalize the resulting coefficient into [0,m).
//
// COMPLEXITY
// O(log m) arithmetic steps and O(1) auxiliary space.
//
// CONTRACT
// 0 <= a < modulus, 2 <= modulus <= 10000. Returns -1 if no inverse exists. This bounded
// integer implementation is not constant time or suitable for cryptographic secrets.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 4.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int modular_inverse(int a, int modulus) {
    alg_require(((modulus >= 2) && (modulus <= 10000) && (a >= 0) && (a < modulus)));
    int old_r = modulus;
    int r = a;
    int old_t = 0;
    int t = 1;
    int quotient = 0;
    int temp = 0;
    while ((r != 0)) {
        quotient = (old_r / r);
        temp = alg_sub(old_r, alg_mul(quotient, r));
        old_r = r;
        r = temp;
        temp = alg_sub(old_t, alg_mul(quotient, t));
        old_t = t;
        t = temp;
    }
    if ((old_r != 1)) {
        return alg_sub(0, 1);
    }
    return (alg_add((old_t % modulus), modulus) % modulus);
}

int main(void) {
    int a = 3;
    int modulus = 11;
    int result = modular_inverse(a, modulus);
    printf("%d\n", result);
    return 0;
}

using System;
// Extended Euclid Modular Inverse in C#
//
// PURPOSE
// Extended Euclid tracks a Bezout coefficient while computing the gcd. When gcd(a,m)=1, that
// coefficient is the modular inverse.
//
// METHOD
// Maintain consecutive remainders and coefficients. Update both with the Euclidean quotient;
// normalize the resulting coefficient into [0,m).
//
// COMPLEXITY
// O(log m) arithmetic steps and O(1) auxiliary space.
//
// CONTRACT
// 0 <= a < modulus, 2 <= modulus <= 10000. Returns -1 if no inverse exists. This bounded
// integer implementation is not constant time or suitable for cryptographic secrets.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 4.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int modular_inverse(int a, int modulus) {
        checked {
            alg_require(((modulus >= 2) && (modulus <= 10000) && (a >= 0) && (a < modulus)));
            int old_r = modulus;
            int r = a;
            int old_t = 0;
            int t = 1;
            int quotient = 0;
            int temp = 0;
            while ((r != 0)) {
                quotient = (old_r / r);
                temp = (old_r - (quotient * r));
                old_r = r;
                r = temp;
                temp = (old_t - (quotient * t));
                old_t = t;
                t = temp;
            }
            if ((old_r != 1)) {
                return (-1);
            }
            return (((old_t % modulus) + modulus) % modulus);
        }
    }

    static void Main() {
        int a = 3;
        int modulus = 11;
        int result = modular_inverse(a, modulus);
        Console.WriteLine(result);
    }
}