Linear Congruence method for generating Pseudo Random Numbers

Last Updated : 11 May, 2021

Linear Congruential Method is a class of Pseudo Random Number Generator (PRNG) algorithms used for generating sequences of random-like numbers in a specific range. This method can be defined as:

$X_{i + 1} = aX_{i} + c \hspace{0.2cm} mod \hspace{0.2cm} m$

where,

X, is the sequence of pseudo-random numbers
m, ( > 0) the modulus
a, (0, m) the multiplier
c, (0, m) the increment
X₀_, [0, m) – Initial value of sequence known as seed

m, a, c, and X₀should be chosen appropriately to get a period almost equal to m.

For a = 1, it will be additive congruence method.
For c = 0, it will be the multiplicative congruence method.

Approach:

Choose the seed value X_0,Modulus parameter m, Multiplier term a, and increment term c.
Initialize the required amount of random numbers to generate (say, an integer variable noOfRandomNums).
Define a storage to keep the generated random numbers (here, vector is considered) of size noOfRandomNums.
Initialize the 0^thindex of the vector with the seed value.
For rest of the indexes follow the Linear Congruential Method to generate the random numbers.

randomNums[i] = ((randomNums[i – 1] * a) + c) % m

Finally, return the random numbers.
Below is the implementation of the above approach:

C++

// C++ implementation of the
// above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to generate random numbers
void linearCongruentialMethod(
    int Xo, int m, int a, int c,
    vector<int>& randomNums,
    int noOfRandomNums)
{
 
    // Initialize the seed state
    randomNums[0] = Xo;
 
    // Traverse to generate required
    // numbers of random numbers
    for (int i = 1; i < noOfRandomNums; i++) {
        // Follow the linear congruential method
        randomNums[i]
            = ((randomNums[i - 1] * a) + c) % m;
    }
}
 
// Driver Code
int main()
{
    int Xo = 5; // Seed value
    int m = 7; // Modulus parameter
    int a = 3; // Multiplier term
    int c = 3; // Increment term
 
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
 
    // To store random numbers
    vector<int> randomNums(
        noOfRandomNums);
 
    // Function Call
    linearCongruentialMethod(
        Xo, m, a, c,
        randomNums, noOfRandomNums);
 
    // Print the generated random numbers
    for (int i = 0; i < noOfRandomNums; i++) {
        cout << randomNums[i] << " ";
    }
 
    return 0;
}

Java

// Java implementation of the above appraoch
import java.util.*;
 
class GFG{
 
// Function to generate random numbers
static void linearCongruentialMethod(int Xo, int m,
                                     int a, int c,
                                     int[] randomNums,
                                     int noOfRandomNums)
{
     
    // Initialize the seed state
    randomNums[0] = Xo;
 
    // Traverse to generate required
    // numbers of random numbers
    for(int i = 1; i < noOfRandomNums; i++)
    {
         
        // Follow the linear congruential method
        randomNums[i] = ((randomNums[i - 1] * a) + c) % m;
    }
}
 
// Driver code
public static void main(String[] args)
{
     
    // Seed value
    int Xo = 5;
     
    // Modulus parameter
    int m = 7;
     
    // Multiplier term
    int a = 3;
     
    // Increment term
    int c = 3;
     
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
     
    // To store random numbers
    int[] randomNums = new int[noOfRandomNums];
     
    // Function Call
    linearCongruentialMethod(Xo, m, a, c,
                             randomNums,
                             noOfRandomNums);
     
    // Print the generated random numbers
    for(int i = 0; i < noOfRandomNums; i++)
    {
        System.out.print(randomNums[i] + " ");
    }
}
}
 
// This code is contributed by offbeat

Python3

# Python3 implementation of the
# above approach
 
# Function to generate random numbers
def linearCongruentialMethod(Xo, m, a, c,
                             randomNums,
                             noOfRandomNums):
 
    # Initialize the seed state
    randomNums[0] = Xo
 
    # Traverse to generate required
    # numbers of random numbers
    for i in range(1, noOfRandomNums):
         
        # Follow the linear congruential method
        randomNums[i] = ((randomNums[i - 1] * a) +
                                         c) % m
 
# Driver Code
if __name__ == '__main__':
     
    # Seed value
    Xo = 5
     
    # Modulus parameter
    m = 7
     
    # Multiplier term
    a = 3
     
    # Increment term
    c = 3
 
    # Number of Random numbers
    # to be generated
    noOfRandomNums = 10
 
    # To store random numbers
    randomNums = [0] * (noOfRandomNums)
 
    # Function Call
    linearCongruentialMethod(Xo, m, a, c,
                             randomNums,
                             noOfRandomNums)
 
    # Print the generated random numbers
    for i in randomNums:
        print(i, end = " ")
 
# This code is contributed by mohit kumar 29

C#

// C# implementation of the above appraoch
using System;
 
class GFG{
 
// Function to generate random numbers
static void linearCongruentialMethod(int Xo, int m,
                                     int a, int c,
                                     int[] randomNums,
                                     int noOfRandomNums)
{
     
    // Initialize the seed state
    randomNums[0] = Xo;
 
    // Traverse to generate required
    // numbers of random numbers
    for(int i = 1; i < noOfRandomNums; i++)
    {
         
        // Follow the linear congruential method
        randomNums[i] = ((randomNums[i - 1] * a) + c) % m;
    }
}
 
// Driver code
public static void Main(String[] args)
{
     
    // Seed value
    int Xo = 5;
     
    // Modulus parameter
    int m = 7;
     
    // Multiplier term
    int a = 3;
     
    // Increment term
    int c = 3;
     
    // Number of Random numbers
    // to be generated
    int noOfRandomNums = 10;
     
    // To store random numbers
    int[] randomNums = new int[noOfRandomNums];
     
    // Function call
    linearCongruentialMethod(Xo, m, a, c,
                             randomNums,
                             noOfRandomNums);
     
    // Print the generated random numbers
    for(int i = 0; i < noOfRandomNums; i++)
    {
        Console.Write(randomNums[i] + " ");
    }
}
}
 
// This code is contributed by sapnasingh4991

Javascript

<script>
 
// Javascript program to implement
// the above approach
 
// Function to generate random numbers
function linearCongruentialMethod(Xo, m,  a, c,
                     randomNums, noOfRandomNums)
{
       
    // Initialize the seed state
    randomNums[0] = Xo;
   
    // Traverse to generate required
    // numbers of random numbers
    for(let i = 1; i < noOfRandomNums; i++)
    {
           
        // Follow the linear congruential method
        randomNums[i] = ((randomNums[i - 1] * a) + c) % m;
    }
}
 
    // Driver Code
         
        // Seed value
    let Xo = 5;
       
    // Modulus parameter
    let m = 7;
       
    // Multiplier term
    let a = 3;
       
    // Increment term
    let c = 3;
       
    // Number of Random numbers
    // to be generated
    let noOfRandomNums = 10;
       
    // To store random numbers
    let randomNums = new Array(noOfRandomNums).fill(0);
       
    // Function Call
    linearCongruentialMethod(Xo, m, a, c,
                             randomNums,
                             noOfRandomNums);
       
    // Print the generated random numbers
    for(let i = 0; i < noOfRandomNums; i++)
    {
        document.write(randomNums[i] + " ");
    }
 
</script>

Output:

5 4 1 6 0 3 5 4 1 6

The literal meaning of pseudo is false. These random numbers are called pseudo because some known arithmetic procedure is utilized to generate. Even the generated sequence forms a pattern hence the generated number seems to be random but may not be truly random.

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