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Chapter 10: Problem 51

Consider an RSA cryptosystem using \(p=23, q=\) 41 and \(g=3 .\) Encipher the message \([847]_{943}\).

Short Answer

Expert verified
The encrypted message using the RSA key pair (943, 3) and the message 847, is obtained by carrying out the encryption \(c = 847^{3} \mod 943 \).

Step by step solution

01

Find the Modulus

The modulus \(n\) is obtained by multiplying the primes \(p\) and \(q\). So, \(n = p \cdot q = 23 \cdot 41 = 943 \)
02

Calculate the Totient

Next, the Euler's totient function \(\phi(n)\) must be calculated, which is \((p-1) \cdot (q-1)\). Here, \(\phi(n) = (23 - 1) \cdot (41 - 1) = 880 \)
03

Determine the public key

The next part of the public key is \(g = 3\), which is already given in the problem. Therefore, the public key is \((n,g) = (943, 3) \)
04

Encipher the message

The final step is to encrypt the message. The given message is [847]_{943}. Using the RSA encryption formula \(c = m^{g} \mod n \), the encrypted message \(c\) is calculated as follows: \(c = 847^{3} \mod 943 \)

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Most popular questions from this chapter

Prove that there are infinitely many prime numbers.

In an RSA cryptosystem, show that if \(\varphi(n)\) can be discovered, then the cryptosystem may be compromised.

show that two integers divide each other if and only if they are equal.

Find all solutions to the equations \([1]_{7} x=[3]_{7}\) and \([12]_{9} x=[6]_{9}\).

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