Encrypting a Secret Code with Exponentiation and Modulo Operations

Encrypting a Secret Code with Exponentiation and Modulo Operations

In this article, we will guide you through a step-by-step approach to help Bob encrypt a secret code using two key values, N and M. The encryption formula involves modular exponentiation and can be computed efficiently, even with large values for N and M.

Problem Overview

Bob wants to send a secret code (S) securely to his boss, but before sending it, he encrypts it using a formula based on two key values: N and M. The formula is as follows:Encrypted Code=(((SN%10)M)%1000000007)\text{Encrypted Code} = (((S^N \% 10)^M) \% 1000000007)Here:
  • SS is the secret code.
  • NN is the first key.
  • MM is the second key.
  • The result is computed modulo 10000000071000000007 (a large prime number used to prevent overflow and ensure secure encryption).

Input Details

The inputs consist of:
  • S (secret code): An integer representing the secret code (1 ≤ S ≤ 10910^9).
  • N (key 1): An integer representing the first key value (0 ≤ N ≤ 101010^{10}).
  • M (key 2): An integer representing the second key value (0 ≤ M ≤ 101010^{10}).

Problem Constraints

The constraints highlight the large size of the numbers involved:
  • S can be up to 10910^9.
  • N and M can be as large as 101010^{10}.
Given these large numbers, we cannot directly compute the powers using naive methods as it may lead to overflow or excessive computation time.

Efficient Solution with Modular Exponentiation

To efficiently compute the result for large values of N and M, we will use the concept of modular exponentiation, which allows us to compute large powers modulo a number in an optimized way. The modular exponentiation technique ensures that we can handle very large exponents without encountering performance issues.

Python Solution

Here is the Python code to encrypt the secret code using the given formula:
python
def mod_exp(base, exp, mod): """Function to perform modular exponentiation: (base^exp) % mod""" result = 1 base = base % mod # Handle base greater than mod while exp > 0: if exp % 2 == 1: # If exponent is odd result = (result * base) % mod exp = exp // 2 # Divide the exponent by 2 base = (base * base) % mod # Square the base return resultdef encrypt_code(S, N, M): “””Encrypt the secret code S using the keys N and M””” # Step 1: Calculate (S^N % 10) S_mod_10 = mod_exp(S, N, 10)# Step 2: Calculate (S_mod_10^M % 1000000007) result = mod_exp(S_mod_10, M, 1000000007)return result# Example usage S = 123456789 N = 10000000000 M = 10000000000 print(encrypt_code(S, N, M)) # Outputs the encrypted code

Explanation of the Code

  1. Modular Exponentiation Function (mod_exp):
    • This function computes baseexp%mod\text{base}^{\text{exp}} \% \text{mod} using an efficient method called exponentiation by squaring.
    • It reduces the number of operations by taking advantage of the binary representation of the exponent. This allows us to handle large exponents in logarithmic time, i.e., O(log⁡(exp))O(\log(\text{exp})).
  2. Encryption Process:
    • First, we calculate SN%10S^N \% 10 to get the last digit of SNS^N. This operation is performed using the mod_exp function with a modulus of 10.
    • Next, we calculate (SN%10)M%1000000007(S^N \% 10)^M \% 1000000007 to get the final encrypted code. This is also done using the mod_exp function with a modulus of 10000000071000000007.
  3. Final Result:
    • The function returns the final encrypted code after performing the above calculations.

Time Complexity

  • The time complexity of the modular exponentiation function is O(log⁡(exp))O(\log(\text{exp})), where exp is the exponent (either N or M).
  • Since we perform two modular exponentiations, the overall time complexity is O(log⁡(N)+log⁡(M))O(\log(N) + \log(M)), which is efficient enough given the problem’s constraints.

Example

For the input:
  • S=123456789S = 123456789
  • N=10000000000N = 10000000000
  • M=10000000000M = 10000000000
The output will be the encrypted code calculated using the formula.

Conclusion

This approach efficiently solves the problem of encrypting a secret code using large exponents and modulo operations. By leveraging modular exponentiation, we avoid overflow and performance issues, even with very large values for N and M. This method ensures that Bob can securely encrypt his secret code for transmission.Click here to know more our program!Encrypting a Secret Code with Exponentiation and Modulo Operations
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