Modular Arithmetic

Modular Arithmetic is a system of arithmetic for integers where numbers wrap around after reaching a certain value called the modulus. It is a crucial concept in various fields, especially in cryptography and number theory.

Core Concepts

Modulus

• The modulus $m$ is the integer at which numbers wrap around.
• For any integer $a$, the expression $a \mod m$ gives the remainder of the division of $a$ by $m$.

Basic Operations

1. Addition : $(a + b) \mod m$
   Example: $(7 + 5) \mod 10 = 2$

   Code:

   C++

   int modular_add(int a, int b, int m) {
       return (a + b) % m;
   }

   Java

   public static int modularAdd(int a, int b, int m) {
    return (a + b) % m;
   }

   Python

   def modular_add(a, b, m):
    return (a + b) % m

   Time Complexity: $O(1)$

2. Subtraction : $(a - b) \mod m$
   Example: $(3 − 4) \mod 5 = 4$

   Code:

   C++

   int modular_sub(int a, int b, int m) {
    return (a - b + m) % m; // ensure non-negative result
   }
   

   Java

   public static int modularSub(int a, int b, int m) {
    return (a - b + m) % m; // ensure non-negative result
    }

   Python

   def modular_sub(a, b, m):
    return (a - b + m) % m  # ensure non-negative result

   Time Complexity: $O(1)$

3. Multiplication:: $(a \times b) \mod m$
   Example: $(4 \times 3) \mod 5 = 2$

   Code:

   C++

   int modular_mul(int a, int b, int m) {
    return (a * b) % m;
    }
   

   Java

   public static int modularMul(int a, int b, int m) {
    return (a * b) % m;
    }
   

   Python

   def modular_mul(a, b, m):
    return (a * b) % m

   Time Complexity: $O(1)$

4. Exponentiation: $(a^b) \mod m$
   Example: $(2^3) \mod 5 = 3$

   Code:

   C++

   int modular_pow(int base, int exp, int mod) {
    int result = 1;
    base = base % mod;
    while (exp > 0) {
        if (exp % 2 == 1) {
            result = (result * base) % mod;
        }
        exp = exp >> 1; // equivalent to exp //= 2
        base = (base * base) % mod;
    }
    return result;
    }
   
   

   Java

   public static int modularPow(int base, int exp, int mod) {
    int result = 1;
    base = base % mod;
    while (exp > 0) {
        if ((exp & 1) == 1) {
            result = (result * base) % mod;
        }
        exp >>= 1; // equivalent to exp /= 2
        base = (base * base) % mod;
    }
    return result;
    }
   
   

   Python

   def modular_pow(base, exp, mod):
    result = 1
    base = base % mod
    while exp > 0:
        if (exp % 2) == 1:
            result = (result * base) % mod
        exp //= 2
        base = (base * base) % mod
    return result
   

   Time Complexity: $O( log \ b)$

Conclusion

Modular arithmetic is a powerful tool in mathematics with significant applications in cryptography, computer science, and number theory. Understanding its core operations and properties is essential for working with modern cryptographic systems and algorithms.