The post "XOR" explores the remarkable versatility of the exclusive-or (XOR) operation in computer programming. It highlights XOR's utility in a variety of contexts, from cryptography (simple ciphers) and data manipulation (swapping variables without temporary storage) to graphics programming (drawing lines and circles) and error detection (parity checks). The author emphasizes XOR's fundamental mathematical properties, like its self-inverting nature (A XOR B XOR B = A) and commutativity, demonstrating how these properties enable elegant and efficient solutions to seemingly complex problems. Ultimately, the post advocates for a deeper appreciation of XOR as a powerful tool in any programmer's arsenal.
This blog post, titled "XOR," delves into the fascinating properties and applications of the exclusive OR (XOR) logical operation. The author begins by establishing the fundamental truth table of XOR, highlighting that it returns true if and only if one of its inputs is true, but not both. This is contrasted with the inclusive OR, which returns true if at least one input is true. The author then meticulously explores the various algebraic identities that XOR adheres to, such as commutativity (A XOR B = B XOR A), associativity (A XOR (B XOR C) = (A XOR B) XOR C), and the self-inverse property (A XOR A = 0). These properties, particularly associativity, are demonstrated through detailed examples and contribute to the elegance and utility of XOR in various computational scenarios.
A core theme of the post is the reversibility of the XOR operation. The author elucidates how XORing a value with a key, and then XORing the result again with the same key, recovers the original value. This characteristic makes XOR exceptionally useful for cryptography, where simple encryption and decryption can be achieved through this "key" based operation. The author further elaborates on this by illustrating a hypothetical scenario of transmitting a secret message. In this scenario, two parties share a secret key beforehand. The sender XORs the message with the key, producing an encrypted ciphertext. The receiver, upon receiving the ciphertext, XORs it with the same shared secret key, perfectly reconstructing the original message. This straightforward example demonstrates the practical power of XOR in secure communication.
Furthermore, the post explores how XOR functions as a bitwise operator in computer programming, affecting individual bits within a binary representation. This bitwise operation is demonstrated with numerical examples, further clarifying its behavior in a computational context. The author concludes by briefly touching upon the applicability of XOR in more complex algorithms, such as RAID 5 parity generation and error detection schemes, where the properties of XOR enable efficient data redundancy and integrity checking. In essence, the post presents a comprehensive overview of XOR, spanning its logical definition, algebraic properties, cryptographic applications, and bitwise operation, emphasizing its elegance and versatile nature in various domains of computer science.
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https://news.ycombinator.com/item?id=43087944
HN users discuss various applications and interpretations of XOR. Some highlight its reversibility and use in cryptography, while others explain its role in parity checks and error detection. A few comments delve into its connection with addition and subtraction in binary arithmetic. The thread also explores the efficiency of XOR in comparison to other bitwise operations and its utility in situations requiring toggling, such as graphics programming. Some users share personal anecdotes of using XOR for tasks like swapping variables without temporary storage. A recurring theme is the elegance and simplicity of XOR, despite its power and versatility.
The Hacker News post titled "XOR" links to an article explaining the XOR (exclusive or) operation. The comments section contains a lively discussion about various aspects of XOR, its uses, and its significance.
Several commenters discuss practical applications of XOR. One commenter highlights its use in cryptography, particularly in simple ciphers and checksums, due to its reversible nature. Another points out its efficiency in RAID systems for parity calculation and data recovery. A different commenter mentions its utility in embedded systems for toggling bits, as well as in graphics programming for drawing lines and implementing collision detection. Someone else mentions its role in certain error-correcting codes, highlighting its mathematical properties.
A few commenters delve into the mathematical properties of XOR, describing it as addition modulo 2, and linking it to concepts like linear independence and vector spaces over GF(2). One commenter explains how XOR forms a group under the operation, where every element is its own inverse.
The elegance and simplicity of XOR are also appreciated by several commenters. One remarks on how a simple operation like XOR can have such wide-ranging applications. Another describes XOR as a "fundamental building block" in computer science.
Some commenters share anecdotes and experiences related to XOR. One recalls learning about XOR through a programming challenge involving swapping two variables without temporary storage. Another shares an example of using XOR in assembly language for efficient bit manipulation.
There's a brief discussion about the difference between logical and bitwise XOR, clarifying their applicability based on the context. One commenter also points out potential confusion arising from different representations of XOR (^, ⊕).
Finally, a few commenters provide additional resources and links to further reading on XOR and related topics, including Wikipedia and other online articles. Overall, the comment section provides a multifaceted perspective on XOR, showcasing its importance and relevance in various fields.