Differential equations of addition

In cryptography, differential equations of addition (DEA) are one of the most basic equations related to differential cryptanalysis that mix additions over two different groups (e.g. addition modulo 2³² and addition over GF(2)) and where input and output differences are expressed as XORs.

Examples of Differential Equations of Addition

Differential equations of addition (DEA) are of the following form:

$(x+y)\oplus ((x\oplus a)+(y\oplus b))=c$

where $x$ and $y$ are $n$ -bit unknown variables and $a$ , $b$ and $c$ are known variables. The symbols $+$ and $\oplus$ denote addition modulo $2^{n}$ and bitwise exclusive-or respectively. The above equation is denoted by $(a,b,c)$ .

Let a set $S=\{(a_{i},b_{i},c_{i})|i$ is an integer less than $k\}$ denote a system of $k$ DEA where $k$ is a polynomial in $n$ . It has been proved that the satisfiability of an arbitrary set of DEA is in the complexity class P when a brute force search requires an exponential time. In 2013, some properties of a special form of

DEA were reported by Chengqing Li et al., where $a=0$ and $y$ is assumed known. Essentially, the special DEA can be represented as $(x\dotplus \alpha )\oplus (x\dotplus \beta )=c$ . Based on the found properties, a algorithm for deriving $x$ was proposed and analyzed.^[1]

Usage of Differential Equations of Addition

Solution to an arbitrary set of DEA (either in batch and or in adaptive query model) was due to Souradyuti Paul and Bart Preneel. The solution techniques have been used to attack the stream cipher Helix.

References

Souradyuti Paul and Bart Preneel, Solving Systems of Differential Equations of Addition, ACISP 2005. Full version (PDF)
Souradyuti Paul and Bart Preneel, Near Optimal Algorithms for Solving Differential Equations of Addition With Batch Queries, Indocrypt 2005. Full version (PDF)
Helger Lipmaa, Johan Wallén, Philippe Dumas: On the Additive Differential Probability of Exclusive-Or. FSE 2004: 317-331.
1. ↑ Li, Chengqing; Liu, Yuansheng; Zhang, Leo Yu; Chen, Michael Z. Q. (2013-04-01). "Breaking a chaotic image encryption algorithm based on modulo addition and xor operation". International Journal of Bifurcation and Chaos. 23 (04): 1350075. doi:10.1142/S0218127413500752. ISSN 0218-1274.

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

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