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Chapter 10: Problem 53

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

Short Answer

Expert verified
The discovery of \( \varphi(n) \) in an RSA cryptosystem can lead to the compromise of the cryptosystem since it allows for the discovery of the prime factors \( p \) and \( q \) of \( n \), which in turn makes it possible to derive the private key from the public key.

Step by step solution

01

Start from the basics of RSA Cryptosystem

In the RSA cryptosystem, we start with selecting two distinct prime numbers \( p \) and \( q \). Their product \( n = pq \) is used as the modulus for both the public and private keys. Our task is based on this modulus \( n \). The totient function \( \varphi(n) \) is calculated as \( \varphi(n) = (p - 1)(q - 1) \).
02

Understand the relationship between \(\varphi(n)\) and n

The Euler's totient function \( \varphi(n) \) reveals how many numbers are less than \( n \) and relatively prime to \( n \). It can be used to find out critical information about the numbers \( p \) and \( q \), which would otherwise remain encrypted by the modulus \( n \). The values \( p \) and \( q \) are crucial as they can be used to derive the private key from the public key.
03

Illustrate the compromise of the RSA cryptosystem

Discovering \( \varphi(n) \) can lead to the decryption of the private key. This is because knowing \( \varphi(n) \) gives the quantities \( (p - 1) \) and \( (q - 1) \). Adding 1 to \( \varphi(n) \) gives the sum \( p + q \). Together with \( n = pq \), these can be viewed as a system of equations with \( p \) and \( q \) as the variables, which can then be solved to find the exact values of \( p \) and \( q \). Once \( p \) and \( q \) are discovered, the private key can be easily obtained. Therefore, if \( \varphi(n) \) can be discovered, the cryptosystem may be compromised.

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