In a RSA cryptosystem, a participant A uses two prime numbers $p = 13$ and $q =17$ to generate his public and private keys. If the public key of a is 41, then the private key of A is_________.
To find the private key in an RSA cryptosystem, we need to first understand how the keys are generated and the relationship between the public and private keys. Given the information:
- Prime numbers: \(p = 13\) and \(q = 17\).
- Public key \(e = 41\).
Here are the steps to find the private key:
- First, calculate \(n\) by multiplying the two prime numbers:
- \(n = p \times q = 13 \times 17 = 221\).
- Calculate Euler's totient function, \(\phi(n)\):
- \(\phi(n) = (p-1)(q-1) = (13-1)(17-1) = 12 \times 16 = 192\).
- The public key, \(e\), must be coprime with \(\phi(n)\). Here, \(e = 41\) and it is coprime with \(\phi(n) = 192\).
- To find the private key, \(d\), we need to find the modular multiplicative inverse of \(e\) modulo \(\phi(n)\). This means finding \(d\) such that:
- \(e \times d \equiv 1 \pmod{\phi(n)}\)
- Solving \(41 \times d \equiv 1 \pmod{192}\), we can use the Extended Euclidean Algorithm:
- Applying the Extended Euclidean Algorithm:
- Use successive divisions to express 1 as a combination of 41 and 192. The calculations give us \(d = 89\).
Thus, the private key of A is \(89\).
Therefore, the correct option is \(89\).
| Option | Private Key |
|---|---|
| 85 | Incorrect |
| 87 | Incorrect |
| 89 | Correct |
| 88 | Incorrect |
Important Questions from Cryptography - Teaching
- In symmetric key cryptography, if every person in a group of 12 people needs to communicate with every other person in the group, how many secret keys are required?
- Match the following:
Match correctly the cryptographic algorithm with their corresponding key-length.1. DES a. 128-256 bits 2. Rj indeal b. 1-448 bits 3. RC4 c. 56 bits 4. Blowfish d. 1-2048 bits - Decryption of the ciphertext "Z" with the key pair (7, 2) in modulus 26 is
In the RSA public key cryptosystem, the private and public keys are $(e, n)$ and $(d, n)$ respectively, where $n = p \times q$ and $p$ and $q$ are large primes. Besides, $n$ is public and $p$ and $q$ are private. Let M be an integer such that $0 <M< n$ and $\phi(n) = (p-1)(q - 1)$. now consider the following equations.
I. M’ =\( M^e\) mod n M =\( (M')^d\) mod nII. ed ≡1 mod n
III. ed ≡1 mod f(n)
IV. M’ = \(M^e \)mod f(n)
M = \( (M')^d\) mod
Which of the above equations correctly represent RSA cryptosystem?- Let M be a message consisting of 50 AES blocks. Alice encrypts using CBC mode and transmits the resulting ciphertext to Bob. Due to a network error, ciphertext block number 45 is corrupted during transmission. All other ciphertext blocks are transmitted and received correctly. Once Bob decrypts the received ciphertext, number of plaintext blocks corrupted is
