Correlation immunity

In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in $x_{1},x_{2},\ldots ,x_{n}$ is statistically independent of the value of $f(x_{1},x_{2},\ldots ,x_{n})$ .

Definition

A function $f:{\mathbb {F}}_{2}^{n}\rightarrow {\mathbb {F}}_{2}$ is $k$ -th order correlation immune if for any independent $n$ binary random variables $X_{0}\ldots X_{{n-1}}$ , the random variable $Z=f(X_{0},\ldots ,X_{{n-1}})$ is independent from any random vector $(X_{{i_{1}}}\ldots X_{{i_{k}}})$ with $0\leq i_{1}<\ldots <i_{k}<n$ .

Results in cryptography

When used in a stream cipher as a combining function for linear feedback shift registers, a Boolean function with low-order correlation-immunity is more susceptible to a correlation attack than a function with correlation immunity of high order.

Siegenthaler showed that the correlation immunity m of a Boolean function of algebraic degree d of n variables satisfies m + d ≤ n; for a given set of input variables, this means that a high algebraic degree will restrict the maximum possible correlation immunity. Furthermore, if the function is balanced then m + d ≤ n − 1.^[1]

References

↑ T. Siegenthaler (September 1984). "Correlation-Immunity of Nonlinear Combining Functions for Cryptographic Applications". IEEE Transactions on Information Theory. 30 (5): 776–780. doi:10.1109/TIT.1984.1056949.

Block ciphers (security summary)

Common algorithms	AES Blowfish DES (Internal Mechanics, Triple DES) Serpent Twofish

Less common algorithms	Camellia CAST-128 IDEA RC2 RC5 RC6 SEED ARIA Skipjack TEA XTEA

Other algorithms	3-Way Akelarre Anubis BaseKing BassOmatic BATON BEAR and LION CAST-256 Chiasmus CIKS-1 CIPHERUNICORN-A CIPHERUNICORN-E CLEFIA CMEA Cobra COCONUT98 Crab Cryptomeria/C2 CRYPTON CS-Cipher DEAL DES-X DFC E2 FEAL FEA-M FROG G-DES GOST Grand Cru Hasty Pudding cipher Hierocrypt ICE IDEA NXT Intel Cascade Cipher Iraqi KASUMI KeeLoq KHAZAD Khufu and Khafre KN-Cipher Kuznyechik Ladder-DES Libelle LOKI (97, 89/91) Lucifer M6 M8 MacGuffin Madryga MAGENTA MARS Mercy MESH MISTY1 MMB MULTI2 MultiSwap New Data Seal NewDES Nimbus NOEKEON NUSH PRESENT Prince Q RC6 REDOC Red Pike S-1 SAFER SAVILLE SC2000 SHACAL SHARK Simon SM4 Speck Spectr-H64 Square SXAL/MBAL Threefish Treyfer UES Xenon xmx XXTEA Zodiac

Design	Feistel network Key schedule Lai-Massey scheme Product cipher S-box P-box SPN Avalanche effect Block size Key size Key whitening (Whitening transformation)

Attack (cryptanalysis)	Brute-force (EFF DES cracker) MITM (Biclique attack, 3-subset MITM attack) Linear (Piling-up lemma) Differential (Impossible Truncated Higher-order) Differential-linear Distinguishing (Known-key) Integral/Square Boomerang Mod n Related-key Slide Rotational Timing XSL Interpolation Partitioning Davies' Rebound Weak key Tau Chi-square Time/memory/data tradeoff

Standardization	AES process CRYPTREC NESSIE

Utilization	Initialization vector Mode of operation Padding

Cryptographic hash functions & message authentication codes

List Comparison Known attacks

Common functions	MD5 SHA-1 SHA-2 SHA-3 BLAKE2

SHA-3 finalists	BLAKE Grøstl JH Skein Keccak (winner)

Other functions	ECOH FSB GOST HAS-160 HAVAL Kupyna LM hash MD2 MD4 MD5 MD6 MDC-2 N-Hash RIPEMD RadioGatún SWIFFT SipHash Snefru Streebog Tiger VSH WHIRLPOOL

Key derivation functions	bcrypt crypt PBKDF2 scrypt Argon2

MAC functions	DAA CBC-MAC HMAC OMAC/CMAC PMAC VMAC UMAC Poly1305

Authenticated encryption modes	CCM CWC EAX GCM IAPM OCB

Attacks	Collision attack Preimage attack Birthday attack Brute force attack Rainbow table Side-channel attack Length extension attack

Design	Avalanche effect Hash collision Merkle–Damgård construction

Standardization	CRYPTREC NESSIE NIST hash function competition

Utilization	Hash-based cryptography Key stretching Merkle tree Message authentication Proof of work Salt

Stream ciphers

Widely used ciphers

eSTREAM Portfolio

Software	HC-256 Rabbit Salsa20 SOSEMANUK

Hardware	Grain MICKEY Trivium

Other ciphers

Theory

Attacks

Cryptography

History of cryptography Cryptanalysis Outline of cryptography

Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography

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Correlation immunity

Definition

Results in cryptography

References

Further reading