RSA (Rivest-Shamir-Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. Here's a step-by-step explanation of how RSA works with a suitable example:

Key Generation:

Choose two distinct prime numbers, for example p = 3 and q = 11.

Compute n = pq = 311 = 33. n is the modulus for both the public and private keys.

Compute the totient function φ(n) = (p-1)(q-1) = 210 = 20.

Choose an integer e such that 1 < e < φ(n) and gcd(φ(n), e) = 1; i.e., e and φ(n) are coprime. Let's choose e = 7.

Determine d as d ≡ e^(-1) mod φ(n), i.e., d is the multiplicative inverse of e modulo φ(n). A value of d = 3 satisfies this condition.

Public key is (e, n) => (7, 33) and private key is (d, n) => (3, 33).

Encryption:

Suppose the sender wishes to send a plaintext message m = 2.

The sender uses the recipient's public key (e, n) to encrypt the message as c = m^e mod n.

So, c = 2^7 mod 33 = 29. The ciphertext c = 29 is sent to the recipient.

Decryption:

The recipient uses their private key (d, n) to decrypt the ciphertext as m = c^d mod n.

So, m = 29^3 mod 33 = 2. The plaintext message m = 2 is recovered.

This is a very basic example. In practice, the prime numbers p and q should be very large (typically hundreds of digits long) to provide sufficient security.

Question

RSA (Rivest-Shamir-Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. Here's a step-by-step explanation of how RSA works with a suitable example:

Key Generation:
- Choose two distinct prime numbers, for example p = 3 and q = 11.
- Compute n = pq = 311 = 33. n is the modulus for both the public and private keys.
- Compute the totient function φ(n) = (p-1)(q-1) = 210 = 20.
- Choose an integer e such that 1 < e < φ(n) and gcd(φ(n), e) = 1; i.e., e and φ(n) are coprime. Let's choose e = 7.
- Determine d as d ≡ e^(-1) mod φ(n), i.e., d is the multiplicative inverse of e modulo φ(n). A value of d = 3 satisfies this condition.
- Public key is (e, n) => (7, 33) and private key is (d, n) => (3, 33).
Encryption:
- Suppose the sender wishes to send a plaintext message m = 2.
- The sender uses the recipient's public key (e, n) to encrypt the message as c = m^e mod n.
- So, c = 2^7 mod 33 = 29. The ciphertext c = 29 is sent to the recipient.
Decryption:
- The recipient uses their private key (d, n) to decrypt the ciphertext as m = c^d mod n.
- So, m = 29^3 mod 33 = 2. The plaintext message m = 2 is recovered.

This is a very basic example. In practice, the prime numbers p and q should be very large (typically hundreds of digits long) to provide sufficient security.

This problem has been solved

Knowee AI · Accepted Answer

RSA (Rivest-Shamir-Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. Here's a step-by-step explanation of how RSA works with a suitable example:

Key Generation:
- Choose two distinct prime numbers, for example p = 3 and q = 11.
- Compute n = pq = 311 = 33. n is the modulus for both the public and private keys.
- Compute the totient function φ(n) = (p-1)(q-1) = 210 = 20.
- Choose an integer e such that 1 < e < φ(n) and gcd(φ(n), e) = 1; i.e., e and φ(n) are coprime. Let's choose e = 7.
- Determine d as d ≡ e^(-1) mod φ(n), i.e., d is the multiplicative inverse of e modulo φ(n). A value of d = 3 satisfies this condition.
- Public key is (e, n) => (7, 33) and private key is (d, n) => (3, 33).
Encryption:
- Suppose the sender wishes to send a plaintext message m = 2.
- The sender uses the recipient's public key (e, n) to encrypt the message as c = m^e mod n.
- So, c = 2^7 mod 33 = 29. The ciphertext c = 29 is sent to the recipient.
Decryption:
- The recipient uses their private key (d, n) to decrypt the ciphertext as m = c^d mod n.
- So, m = 29^3 mod 33 = 2. The plaintext message m = 2 is recovered.

This is a very basic example. In practice, the prime numbers p and q should be very large (typically hundreds of digits long) to provide sufficient security.

This problem has been solved

Explain RSA with suitable example.

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