Question

Suppose two people want to play poker over the network. To "deal" the cards they need a mechanism for fairly choosing a random number \(x\) between them; each party stands to lose if the other party can unfairly influence the choice of \(x\). Describe such a mechanism. Hint: You may assume that if either of two bit strings \(x_{1}\) and \(x_{2}\) are random, then the exclusive-OR \(x=x_{1} \oplus x_{2}\) is random.

Short answer

Generate random bit strings, exchange them, compute the XOR, and use the result as the random number.

Step-by-step solution

Generate Random Bit Strings

Both players should independently generate a random bit string. Let's call these bit strings \( x_{1} \) and \( x_{2} \). These strings need to be securely generated so that neither player can predict the other’s string.

Exchange the Bit Strings

The two players should then securely exchange their respective bit strings over the network. This can be done using encryption to ensure security and prevent interception or tampering.

Compute the XOR

Once both bit strings have been exchanged, each player computes the exclusive-OR (XOR) of these two strings. The formula is \( x = x_{1} \bigoplus x_{2} \). This operation will provide a new bit string, \( x \), based on both initial strings.

Use the Result

The resultant bit string \( x \) is the random number that will be used. Each player can now use \( x \) knowing that it was fairly generated and neither player could single-handedly influence its outcome.

Key concepts

XOR Operation

The XOR operation, also known as 'exclusive OR,' is a fundamental concept in computer science and digital logic. This operation takes two input bits and returns one output bit based on a simple rule: the output is true if and only if the inputs are different.
For instance, when you XOR two bits: 0 XOR 0 = 0, 0 XOR 1 = 1, 1 XOR 0 = 1, and 1 XOR 1 = 0. Why is XOR useful in generating random bit strings? It ensures that if either of two input strings is random, the resulting string remains random. This is why it is often used in cryptographic protocols.
The XOR operation can also be performed on longer bit strings by applying the XOR operation bit by bit across the strings. For example, given two bit strings 1101 and 1011, the XOR result would be 0110. This property makes XOR a great tool for fair random number generation in secure communications.
The ability to provide randomness and unrecoverable outcomes from known inputs makes XOR a powerful tool for ensuring fairness and mitigating bias during random number generation. It's essential in both cryptography and network security.

Random Bit String

A random bit string is a sequence of bits (0s and 1s) generated in such a way that each bit is equally likely to be 0 or 1, independently of other bits. Random bit strings are crucial for various applications in cryptography, gaming, and secure communications. To generate a random bit string, it's important to use a secure random number generator (RNG). A secure RNG ensures that each bit in the generated string is unpredictably random and not influenced by any external factors.
In our context, both players independently generate a random bit string. These strings need to be long enough to ensure unpredictability and security. For example, generating a 128-bit string provides a high level of randomness. Once generated, the security of these bit strings is maintained by preventing any external party from predicting them.
Using these independently generated random bit strings, the XOR operation ensures that the final outcome remains random as long as at least one of the two strings was genuinely random. This characteristic is particularly important in network communications where fairness and security are paramount, like in the poker game scenario described in the original exercise.

Network Communication Security

When players exchange their random bit strings over a network, they must do so securely to prevent any third parties from intercepting or tampering with the data. Network communication security involves encrypting the data being sent so that only the intended recipient can decode and use it.
Encryption is a process where plain text is converted into a coded form using an encryption algorithm and a key. This ensures that even if the data is intercepted, it cannot be read or altered by anyone who does not have the corresponding decryption key. There are various encryption techniques like symmetric and asymmetric encryption. For simple and secure exchange, players can use methods like Transport Layer Security (TLS) or Advanced Encryption Standard (AES) to encrypt their bit strings.
Beyond encryption, network communication security also involves verifying the identity of the communicating parties, ensuring data integrity, and protecting against attacks like eavesdropping, man-in-the-middle, and replay attacks. Implementing these measures ensures that the bit strings remain confidential and unaltered during transmission, maintaining the fairness and randomness required for generating the final random number for the poker game.