Secret Keys for N People: Calculation
To ensure privacy using secret key encryption between N people, each person needs a distinct secret key shared only with one other specific person. This means we need to find the number of unique pairs that can be formed from N individuals.
Deriving the Number of Secret Keys
The problem boils down to selecting groups of 2 people from a total of N people, where the order of selection does not matter. This is a classic combination problem.
The formula for combinations is:
$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $
Here, $n$ represents the total number of people ($N$), and $k$ represents the size of the group for each secret key (which is 2, as a key is shared between a pair).
Substituting $n = N$ and $k = 2$ into the formula:
Number of keys = $ \binom{N}{2} = \frac{N!}{2!(N-2)!} $
Now, we simplify the expression:
Number of keys = $ \frac{N \times (N-1) \times (N-2)!}{2 \times 1 \times (N-2)!} $
Canceling out $(N-2)!$ from the numerator and denominator:
Number of keys = $ \frac{N(N - 1)}{2} $
Conclusion
Therefore, the total number of secret keys required for N people to use secret key encryption for privacy is $ \frac{N(N - 1)}{2} $.
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
