Question

The Diffie-Hellman key exchange is being used to establish a secret key between Alice and Bob. Alice sends Bob \((227,5,82)\). Bob responds with (125). Alice's secret number, \(x\), is 12 , and Bob's secret number, \(y\), is 3 . Show how Alice and Bob compute the secret key.

Short answer

Alice and Bob's shared secret key is 196.

Step-by-step solution

Identify the variables

Alice sends Bob the tuple (227,5,82), where 227 is the prime number \( p \), 5 is the primitive root \( g \), and 82 is the computed value \( A \). Bob sends Alice his computed value which is 125.

Alice computes her public value

Alice computes her public value \( A = g^x \mod p \). With Alice's secret number \( x = 12 \), the calculation is \( A = 5^{12} \mod 227 = 82 \). This value was sent to Bob.

Bob computes his public value

Bob computes his public value \( B = g^y \mod p \). With Bob's secret number \( y = 3 \), the calculation is \( B = 5^3 \mod 227 = 125 \). This value was sent to Alice.

Alice computes the shared secret key

Alice uses Bob's public value to compute the shared secret key \( K = B^x \mod p \). Substituting the known values results in \( K = 125^{12} \mod 227 \). Calculating this gives \( K = 196 \).

Bob computes the shared secret key

Bob uses Alice's public value to compute the shared secret key \( K = A^y \mod p \). With the given values, \( K = 82^3 \mod 227 \). Calculating this gives \( K = 196 \).

Key concepts

Cryptography

Cryptography is a crucial field in computer science and mathematics focused on protecting information and communications through the use of codes. This ensures that only those for whom the information is intended can read and process it. It's an essential aspect of securing data in the modern digital age.

• It involves techniques such as encryption and decryption, designed to encode messages, ensuring privacy.
• Though widely used today in online communications, its history spans back to ancient times with simple ciphers.
• In today's context, cryptography enables secure online transactions, protecting sensitive data from unauthorized access.

In the context of the Diffie-Hellman key exchange, cryptography allows two parties, like Alice and Bob, to agree on a shared secret over an unsecured communication channel. They do this without needing to share the key beforehand. This agreement relies heavily on mathematical principles, ensuring that even if someone intercepts their messages, the underlying secret remains confidential.

Public Key Exchange

Public key exchange is a fundamental principle in modern cryptography systems, allowing users to share private keys over a public channel securely. It's the underlying method for ensuring that parties such as Alice and Bob can securely exchange data without sharing sensitive keys openly.

• This method uses a pair of keys: a public key, which is shared, and a private key, which remains confidential.
• In the Diffie-Hellman key exchange scenario, Alice and Bob use their private numbers and publicly share calculations derived from these numbers to compute a shared secret.
• It enables secure communication, whereby knowledge of the public key alone does not allow the derivation of the private key or the shared secret.

The brilliance of public key exchange lies in its ability to harness mathematical challenges that ensure security. Even if a third party captures all messages exchanged during the key exchange process, they cannot feasibly compute the shared secret.

Modular Arithmetic

Modular arithmetic is a mathematical system used in cryptography that deals with numbers wrapping around upon reaching a certain value, called the modulus. It's crucial for cryptographic algorithms like Diffie-Hellman, where computations need to stay within a feasible range.

• In modular arithmetic, numbers wrap around after reaching the modulus, much like a clock resets after 12.
• This wrapping effect is utilized in computing values that need to fit within specific boundaries, avoiding overflow or large numbers.
• In the case of Diffie-Hellman, Alice and Bob compute values using a shared prime modulus to maintain computational integrity and confidentiality.

When Alice computes her public value as part of the Diffie-Hellman exchange, she does so using the formula \(A = g^x \mod p\). Here, \(g\) is the base, \(x\) is her secret number, and \(p\) is the prime modulus. The output is a manageable, encrypted number that Bob can also manipulate with his numbers to agree upon the secret key. This cycle of computation underpins the strength of the Diffie-Hellman protocol.