XOR cipher

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

:

And conversely, for decryption:

:

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.{{Harvnb|Churchhouse|2002|p=68}} 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 $\backslash oplus$ ciphertext = key.

It is also trivial to flip arbitrary bits in the decrypted plaintext by manipulating the ciphertext.

This is called malleability.

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.{{sfn | Paar | Pelzl | 2009|pp=32–34}}

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.{{sfn | Paar | Pelzl | 2009|pp=32–34}}

Other logical operations such and AND or OR do not have such a mapping (for example, AND would produce three 0's and one 1, so knowing that a given ciphertext bit is a 0 implies that there is a 2/3 chance that the original plaintext bit was a 0, as opposed to the ideal 1/2 chance in the case of XOR){{efn|name=3ways}}

Example using the Python programming language.{{efn|name=inspired}}

from os import urandom

def generate_key(length: int) -> bytes:

"""Generate encryption key."""

return urandom(length)

def xor_strings(s, t) -> bytes:

"""Concatenate xor two strings together."""

if isinstance(s, str):

# Text strings contain single characters

return "".join(chr(ord(a) ^ b) for a, b in zip(s, t)).encode("utf8")

else:

# Bytes objects contain integer values in the range 0-255

return bytes([a ^ b for a, b in zip(s, t)])

message = "This is a secret message"

print("Message:", message)

key = generate_key(len(message))

print("Key:", key)

cipherText = xor_strings(message.encode("utf8"), key)

print("cipherText:", cipherText)

print("decrypted:", xor_strings(cipherText, key).decode("utf8"))

1. Verify

if xor_strings(cipherText, key).decode("utf8") == message:

print("Unit test passed")

else:

print("Unit test failed")

A shorter example using the R programming language, based on a [https://www.instagram.com/p/CfcTwYYlnC9/ puzzle] posted on Instagram by GCHQ.

secret_key <- c(0xc6, 0xb5, 0xca, 0x01) |> as.raw()

secret_message <- "I <3 Wikipedia" |>

charToRaw() |>

xor(secret_key) |>

base64enc::base64encode()

secret_message_bytes <- secret_message |>

base64enc::base64decode()

xor(secret_message_bytes, secret_key) |> rawToChar()

• Block cipher
• Vernam cipher
• Vigenère cipher

{{notelist|refs=

{{efn|name=3ways|There are 3 ways of getting a (ciphertext) output bit of 0 from an AND operation:

Plaintext=0, key=0;

Plaintext=0, key=1;

Plaintext=1, key=0.

Therefore, if we know that the ciphertext bit is a 0, there is a 2/3 chance that the plaintext bit was also a 0 for a truly random key. For XOR, there are exactly 2 ways, so the chance is 1/2 (i.e. equally likely, so we cannot learn anything from this information)

}}

{{efn|name=inspired|This was inspired by {{harvnb|Richter|2012}} }}

}}

{{reflist}}

{{refbegin}}

• {{cite journal | last1=Budiman | first1=MA | last2=Tarigan | first2=JT | last3=Winata | first3=AS | title=Arduino UNO and Android Based Digital Lock Using Combination of Vigenere Cipher and XOR Cipher | journal=Journal of Physics: Conference Series | publisher=IOP Publishing | volume=1566 | year=2020 | issue=1 | issn=1742-6588 | doi=10.1088/1742-6596/1566/1/012074 | page=012074| bibcode=2020JPhCS1566a2074B | doi-access=free }}
• {{Citation | last = Churchhouse | first = Robert | author-link = Robert Churchhouse | title = Codes and Ciphers: Julius Caesar, the Enigma and the Internet | place = Cambridge | publisher = Cambridge University Press | year = 2002 | isbn = 978-0-521-00890-7 | url-access = registration | url = https://archive.org/details/codesciphersjuli0000chur }}
• {{cite journal | last=Garg | first=Satish Kumar | title=Cryptography Using Xor Cipher | journal=Research Journal of Science and Technology | publisher=A and V Publications | volume=9 | issue=1 | year=2017 | issn=0975-4393 | doi=10.5958/2349-2988.2017.00004.3 | page=25}}
• {{Cite journal|last=Gödel|first=Kurt|date=December 1931|title=Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I|journal=Monatshefte für Mathematik und Physik|language=de|volume=38-38|issue=1|pages=173–198|doi=10.1007/BF01700692|s2cid=197663120|issn=0026-9255}}
• {{Cite journal |title=All About XOR |last=Lewin |first=Michael |journal=Overload |volume=2 |issue=(109 |pages=14–19 |date=June 2012 |access-date=29 August 2021 |url= https://accu.org/journals/overload/20/109/lewin_1915/ }}
• {{cite book| last1=Paar | first1=Christof | last2=Pelzl | first2=Jan | title=Understanding cryptography : a textbook for students and practitioners | publisher=Springer | year=2009 | isbn=978-3-642-04101-3 | oclc=567365751 }}
• {{Citation | last = Richter| first = Wolfgang | title = Unbreakable Cryptography in 5 Minutes | journal = Crossroads: The ACM Magazine for Students | date = August 3, 2012 | publisher = Association for Computing Machinery | url = http://xrds.acm.org/blog/2012/08/unbreakable-cryptography-in-5-minutes/ }}
• {{Citation | last = Tutte | first = W. T. | author-link = W. T. Tutte | title = Fish and I | date = 19 June 1998 | url = https://uwaterloo.ca/combinatorics-and-optimization/sites/ca.combinatorics-and-optimization/files/uploads/files/corr98-39.pdf | access-date = 11 January 2020 }} Transcript of a lecture given by Prof. Tutte at the University of Waterloo

{{refend}}

Category:Stream ciphers