Piling-up lemma

In cryptanalysis, the piling-up lemma is a principle used in linear cryptanalysis to construct linear approximation to the action of block ciphers. It was introduced by Mitsuru Matsui (1993) as an analytical tool for linear cryptanalysis.

Theory

The piling-up lemma allows the cryptanalyst to determine the probability that the equality:

X_{1}\oplus X_{2}\oplus \cdots \oplus X_{n}=0

holds, where the X 's are binary variables (that is, bits: either 0 or 1).

Let P(A) denote "the probability that A is true". If it equals one, A is certain to happen, and if it equals zero, A cannot happen. First of all, we consider the piling-up lemma for two binary variables, where $P(X_{1}=0)=p_{1}$ and $P(X_{2}=0)=p_{2}$ .

Now, we consider:

P(X_{1}\oplus X_{2}=0)

Due to the properties of the xor operation, this is equivalent to

P(X_{1}=X_{2})

X₁ = X₂ = 0 and X₁ = X₂ = 1 are mutually exclusive events, so we can say

P(X_{1}=X_{2})=P(X_{1}=X_{2}=0)+P(X_{1}=X_{2}=1)=P(X_{1}=0,X_{2}=0)+P(X_{1}=1,X_{2}=1)

Now, we must make the central assumption of the piling-up lemma: the binary variables we are dealing with are independent; that is, the state of one has no effect on the state of any of the others. Thus we can expand the probability function as follows:

$P(X_{1}\oplus X_{2}=0)$	$=P(X_{1}=0)P(X_{2}=0)+P(X_{1}=1)P(X_{2}=1)\$
	$=p_{1}p_{2}+(1-p_{1})(1-p_{2})\$
	$=p_{1}p_{2}+(1-p_{1}-p_{2}+p_{1}p_{2})\$
	$=2p_{1}p_{2}-p_{1}-p_{2}+1\$

Now we express the probabilities p₁ and p₂ as ½ + ε₁ and ½ + ε₂, where the ε's are the probability biases — the amount the probability deviates from ½.

$P(X_{1}\oplus X_{2}=0)$	$=2(1/2+\epsilon _{1})(1/2+\epsilon _{2})-(1/2+\epsilon _{1})-(1/2+\epsilon _{2})+1\$
	$=1/2+\epsilon _{1}+\epsilon _{2}+2\epsilon _{1}\epsilon _{2}-1/2-\epsilon _{1}-1/2-\epsilon _{2}+1\$
	$=1/2+2\epsilon _{1}\epsilon _{2}\$

Thus the probability bias ε_1,2 for the XOR sum above is 2ε₁ε₂.

This formula can be extended to more X 's as follows:

P(X_{1}\oplus X_{2}\oplus \cdots \oplus X_{n}=0)=1/2+2^{{n-1}}\prod _{{i=1}}^{n}\epsilon _{i}

Note that if any of the ε's is zero; that is, one of the binary variables is unbiased, the entire probability function will be unbiased — equal to ½.

A related slightly different definition of the bias is $\epsilon _{i}=P(X_{i}=1)-P(X_{i}=0),$ in fact minus two times the previous value. The advantage is that now with

$\varepsilon _{{total}}=P(X_{1}\oplus X_{2}\oplus \cdots \oplus X_{n}=1)-P(X_{1}\oplus X_{2}\oplus \cdots \oplus X_{n}=0)$

we have

\varepsilon _{{total}}=(-1)^{{n+1}}\prod _{{i=1}}^{n}\varepsilon _{i},

adding random variables amounts to multiplying their (2nd definition) biases.

Practice

In practice, the Xs are approximations to the S-boxes (substitution components) of block ciphers. Typically, X values are inputs to the S-box and Y values are the corresponding outputs. By simply looking at the S-boxes, the cryptanalyst can tell what the probability biases are. The trick is to find combinations of input and output values that have probabilities of zero or one. The closer the approximation is to zero or one, the more helpful the approximation is in linear cryptanalysis.

However, in practice, the binary variables are not independent, as is assumed in the derivation of the piling-up lemma. This consideration has to be kept in mind when applying the lemma; it is not an automatic cryptanalysis formula.

References

Matsui, Mitsuru (1994). "Linear Cryptanalysis Method for DES Cipher". Advances in Cryptology—EUROCRYPT ’93. Springer. pp. 386–397. doi:10.1007/3-540-48285-7_33.

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

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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