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| 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 2<sup>32</sup> and addition over GF(2)) and where input and output differences are expressed as XORs.
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| == Examples of Differential Equations of Addition ==
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| '''Differential equations of addition''' (DEA) are of the following form: | |
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| <math>(x+y)\oplus((x\oplus a)+(y\oplus b))=c</math>
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| where <math>x</math> and <math>y</math> are <math>n</math>-bit '''unknown''' variables and <math>a</math>, <math>b</math> and <math>c</math> are '''known''' variables. The symbols <math>+</math> and <math>\oplus</math> denote ''addition modulo'' <math>2^n</math> and ''bitwise exclusive-or'' respectively. The above equation is denoted by <math>(a, b, c)</math>.
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| Let a set <math>S=\{(a_i, b_i, c_i)|i</math> is an integer less than <math>k\}</math> denote a system of <math>k</math> '''DEA''' where <math>k</math> is a polynomial in <math>n</math>. It has been proved that the satisfiability of an arbitrary set of DEA is in the '''[[P = NP problem|complexity class P]]''' when a brute force search requires an [[exponential time]].
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| == Usage of Differential Equations of Addition ==
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| 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 [[Phelix|Helix]].
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| == References ==
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| * [[Souradyuti Paul]] and [[Bart Preneel]], Solving Systems of Differential Equations of Addition, ACISP 2005. [http://www.cosic.esat.kuleuven.be/publications/article-566.pdf Full version] ([[PDF]])
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| * [[Souradyuti Paul]] and [[Bart Preneel]], Near Optimal Algorithms for Solving Differential Equations of Addition With Batch Queries, [[Indocrypt]] 2005. [http://www.cosic.esat.kuleuven.be/publications/article-587.pdf Full version] ([[PDF]])
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| * Helger Lipmaa, Johan Wallén, Philippe Dumas: On the Additive Differential Probability of Exclusive-Or. [[Fast Software Encryption|FSE]] 2004: 317-331.
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| {{Cryptography navbox | block}}
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| [[Category:Cryptographic attacks]]
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| [[Category:Theory of cryptography]]
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| [[Category:Ciphers]]
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| [[Category:Algebra]]
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The writer is known as Araceli Gulledge. For many years she's been residing in Kansas. Playing croquet is something I will by no means give up. My occupation is a production and distribution officer and I'm doing pretty good financially.
Feel free to visit my webpage: neonex.cl