By Valerie Nachef, Jacques Patarin, Emmanuel Volte

This publication offers a survey on other kinds of Feistel ciphers, with their definitions and mathematical/computational houses. Feistel ciphers are primary in cryptography so as to receive pseudorandom diversifications and secret-key block ciphers. partly 1, we describe Feistel ciphers and their versions. We additionally supply a short tale of those ciphers and easy safeguard effects. partly 2, we describe primary assaults on Feistel ciphers. partially three, we provide effects on DES and particular Feistel ciphers. half four is dedicated to superior safeguard effects. We additionally supply effects on indifferentiability and indistinguishability.

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**Extra info for Feistel Ciphers: Security Proofs and Cryptanalysis**

**Example text**

2 2 2 2 2 Case 3: R1 ¤ R2 ; S1 D S2 and R1 ˚ R2 D T1 ˚ T2 or : R1 D R2 ; SÂ1 ¤ S2 and S1 ˚ S2 D L1 ˚ L2 Ã 1 1 2 1 . r 1/n 2. 2 2/n 2. 2 1/n 2 2 Case 4: R1 D R2 and S1 D S2 Â Ã 1 . r 2/n Proof. This is a sketch of the proof. The main idea is to make an induction on r. Let hi D Hr 24n ; jFn jr and h0i D HrC2 24n ; jFn jrC2 where i, 1 Ä i Ä 4 denotes the case number i. Then: 8 0 h ˆ ˆ < 10 h2 ˆ h0 ˆ : 30 h4 D D D D 1 1 1 1 1 2n 1 2n 1 2n 2 2n C 212n h1 C 21n h1 C h22n h1 C h23n h2 C h23n C h24n 2 2n h2 C h3 2n t u When r is odd and r 3, the computation of Hr is also possible.

2. n/ be a polynomial in n. n/ a key. n/ ! Fn such that: 1. ˛/ can be computed in polynomial time in n. 1). 2. n/, and for all polynomials in n distinguisher D, D using a function f W f0; 1gn ! f / ! k// ! n/ To summarize, a pseudo-random function generator is a polynomial technique to obtain a pseudo random function from a random key k. x/ (in a time considered as unity). 3. k/ is 1–1 onto). k/) from a pseudo-random permutation generator. x/. With these definitions, the attacks on Feistel ciphers given in Chap.

These theorems are the basis of a general proof technique called the “H-coefficient technique”, that allows to prove security results for function generators and permutation generators (and thus applies for random and pseudo-random Feistel ciphers). Five of these theorems were mentioned and proved in [7–9]. 3) was proved in [1]. The “Expectation Method” of [2] can also be seen as another related theorem. © Springer International Publishing AG 2017 V. 1 Notation: Definition of H • K will denote a set of values that we will sometimes call “keys”.