science

XOR cipher

XOR cipher is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand XOR cipher rather than just read about it. In short: In cryptography, the simple XOR cipher is a type of additive cipher, an encryption algorithm that operates according to the principles: A ⊕ {\displaystyle \oplus } 0 = A, A ⊕ {\displaystyle \oplus } A = 0, A ⊕ {\displaystyle \oplus } B = B ⊕ {\displaystyle \oplus } A, (A ⊕ {\displaystyle \oplus } B) ⊕ {\displaystyle \oplus } C = A ⊕ {\displaystyle \oplus } (B ⊕ {\displaystyle \oplus } C), (B ⊕ {\displaystyle \oplus…

Key takeaways

XOR cipher belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect XOR cipher to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of XOR cipher from memory before moving on to harder problems.

Reference excerpt

In cryptography, the simple XOR cipher is a type of additive cipher, an encryption algorithm that operates according to the principles:

A ⊕ {\displaystyle \oplus } 0 = A, A ⊕ {\displaystyle \oplus } A = 0, A ⊕ {\displaystyle \oplus } B = B ⊕ {\displaystyle \oplus } A, (A ⊕ {\displaystyle \oplus } B) ⊕ {\displaystyle \oplus } C = A ⊕ {\displaystyle \oplus } (B ⊕ {\displaystyle \oplus } C), (B ⊕ {\displaystyle \oplus } A) ⊕ {\displaystyle \oplus } A = B ⊕ {\displaystyle \oplus } 0 = B For example where ⊕ {\displaystyle \oplus } denotes the exclusive disjunction (XOR) operation. This operation is sometimes called modulus 2 addition (or subtraction, which is identical). With this logic, a string of text can be encrypted by applying the bitwise XOR operator to every character using a given key. To decrypt the output, merely reapplying the XOR function with the key will remove the cipher.

Example The string "Wiki" (01010111 01101001 01101011 01101001 in 8-bit ASCII) can be encrypted with the repeating key 11110011 as follows:

And conversely, for decryption:

Use and security The XOR operator is extremely common as a component in more complex ciphers. By itself, using a constant repeating key, a simple XOR cipher can trivially be broken using frequency analysis. If the content of any message can be guessed or otherwise known then the key can be revealed. Its primary merit is that it is simple to implement, and that the XOR operation is computationally inexpensive. A simple repeating XOR (i.e. using the same key for xor operation on the whole data) cipher is therefore sometimes used for hiding information in cases where no particular security is required. The XOR cipher is often used in computer malware to make reverse engineering more difficult. If the key is random and is at least as long as the message, the XOR cipher is much more secure than when there is key repetition within a message. When the keystream is generated by a pseudo-random number generator, the result is a stream cipher. With a key that is truly random, the result is a one-time pad, which is unbreakable in theory. The XOR operator in any of these ciphers is vulnerable to a known-plaintext attack, since plaintext ⊕ {\displaystyle \oplus } ciphertext = key. It is also trivial to flip arbitrary bits in the decrypted plaintext by manipulating the ciphertext. This is called malleability.

Usefulness in cryptography The primary reason XOR is so useful in cryptography is because it is "perfectly balanced"; for a given plaintext input 0 or 1, the ciphertext result is equally likely to be either 0 or 1 for a truly random key bit. The table below shows all four possible pairs of plaintext and key bits. It is clear that if nothing is known about the key or plaintext, nothing can be determined from the ciphertext alone.

Other logical operations such and AND or OR do not have such a mapping. For example consider the table for AND below:

If the ciphertext were 0, then there is a 2/3 chance that the plaintext was 0 too. And if the ciphertext was 1, then the plaintext would have to be 1. This clearly reveals information about the text that the XOR approach does not.

Example implementation Example using the JavaScript programming language.Another example using the Python programming language.

A shorter example using the R programming language, based on a puzzle posted on Instagram by GCHQ.

See also Block cipher Vernam cipher Vigenère cipher

References

Notes

Citations

Sources

Worked examples

Example 1 — a first encounter with XOR cipher

Start with the simplest possible case. Write down what XOR cipher claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to XOR cipher before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about XOR cipher ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of XOR cipher

In research: XOR cipher appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses XOR cipher in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: XOR cipher is common in secondary-school and first-year university syllabi. It links to neighbouring topics Stream ciphers, so understanding it makes those chapters shorter.
In everyday life: Look for XOR cipher outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “XOR cipher” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study XOR cipher in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what XOR cipher means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain XOR cipher out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is XOR cipher in simple terms?

In cryptography, the simple XOR cipher is a type of additive cipher, an encryption algorithm that operates according to the principles: A ⊕ {\displaystyle \oplus } 0 = A, A ⊕ {\displaystyle \oplus } A = 0, A ⊕ {\displaystyle \oplus } B = B ⊕ {\displaystyle \oplus } A, (A ⊕ {\displaystyle \oplus } B…

Why does XOR cipher matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study XOR cipher?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on XOR cipher.