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Pseudorandom Permutation Families over Abelian Groups [chapter]

Louis Granboulan, Éric Levieil, Gilles Piret
2006 Lecture Notes in Computer Science  
We prove piling-up lemmas for the generalized differential probability and the linear potential, and we study their lower bounds and average value, in particular in the case of permutations of Fp.  ...  Introduction Motivations While all well-known block ciphers are pseudo-random permutation families of some set {0, 1} n where n = 64 or 128, there exists some applications where a pseudo-random permutation  ...  Conclusion In this paper we extended usual block cipher theory over Z n 2 to a more general framework in which the input and output spaces are arbitrary abelian groups.  ... 
doi:10.1007/11799313_5 fatcat:hir3vjvrj5cpvcrsbbuhovpd34

Luby-Racko. Ciphers: Why XOR Is Not So Exclusive [chapter]

Sarvar Patel, Zulfikar Ramzan, Ganpathy S. Sundaram
2003 Lecture Notes in Computer Science  
In some cases, we can break these ciphers over arbitrary Abelian groups and in other cases, however, the security remains an open problem.  ...  We also discuss the existence (and construction) of this function family over various groups, and argue the necessity of this family in our construction.  ...  We cite our main theorem: We remark that although ∆-universal hash functions are traditionally defined over Abelian groups one could easily extend the definition to hold over non-Abelian groups, and our  ... 
doi:10.1007/3-540-36492-7_18 fatcat:4z2tpxzgxbhvrcvx3s4lmlrup4

Small-Bias Spaces for Group Products [chapter]

Raghu Meka, David Zuckerman
2009 Lecture Notes in Computer Science  
Our construction exploits the fact that solvable groups have nontrivial normal subgroups that are abelian and builds on the construction of Azar et al. [AMN98] for abelian groups.  ...  Besides being natural, our extension captures some of the difficulties in constructing pseudorandom generators for constant-width branching programs -a longstanding open problem.  ...  Previous Work and Preliminaries We first present the notions of pseudorandom generators for small width branching programs and small-bias spaces over abelian groups.  ... 
doi:10.1007/978-3-642-03685-9_49 fatcat:3ifjxk7z2jchjb7mzbfgyxdlrm

How to Generate Pseudorandom Permutations Over Other Groups [article]

Hector Bjoljahn Hougaard
2017 arXiv   pre-print
Finally, we consider Zhandry's result on quantum pseudorandom permutations, showing that his result may be generalized to hold for arbitrary groups.  ...  After generalizing the Feistel cipher to arbitrary groups we resolve an open problem of Patel, Ramzan, and Sundaram by showing that the 3-round Feistel cipher over an arbitrary group is not super pseudorandom  ...  scheme over a group G is a super pseudorandom permutation.  ... 
arXiv:1710.05645v1 fatcat:h2kpvqo765dv5dsd3tpkgixjsq

Towards Making Luby-Rackoff Ciphers Optimal and Practical [chapter]

Sarvar Patel, Zulfikar Ramzan, Ganapathy S. Sundaram
1999 Lecture Notes in Computer Science  
The path breaking paper of Luby and Rackoae ë7ë described the construction of pseudorandom permutation generators from pseudorandom function generators, which enabled the formalism of the notion of a block  ...  Pseudorandom permutations can then be interpreted as block ciphers that are secure against adaptive c hosen plaintext and ciphertext attacks.  ...  Let G be an Abelian Group and let 0 , 0 and 0 + 0 denote the subtraction and addition operations with respect to this group.  ... 
doi:10.1007/3-540-48519-8_13 fatcat:7tgtlm7qtvbwnhgobgwxcxg7ou

Quantum-Secure Symmetric-Key Cryptography Based on Hidden Shifts [chapter]

Gorjan Alagic, Alexander Russell
2017 Lecture Notes in Computer Science  
In this work, we study simple algebraic adaptations of such schemes that replace ( Z/2)^n addition with operations over alternate finite groups--such as Z/2^n--and provide evidence that these adaptations  ...  We show that a Hidden Shift version of the Even-Mansour block cipher yields a quantum-secure pseudorandom function, and that a Hidden Shift version of the Encrypted CBC-MAC yields a collision-resistant  ...  Let G be either the Z/2 n group family or the S n group family. Under Assumption 4, the Hidden Shift Even-Mansour cipher over G is a quantum-secure pseudorandom function. Proof.  ... 
doi:10.1007/978-3-319-56617-7_3 fatcat:x7btopqmzbguzlsunujxqfr7be

How to Generate Pseudorandom Permutations Over Other Groups: Even-Mansour and Feistel Revisited [article]

Hector Bjoljahn Hougaard
2017 arXiv   pre-print
After generalizing the Feistel cipher to arbitrary groups we resolve an open problem of Patel, Ramzan, and Sundaram by showing that the 3-round Feistel cipher over an arbitrary group is not super pseudorandom  ...  We generalize the result by Kilian and Rogaway, that the Even-Mansour cipher is pseudorandom, to super pseudorandomness, also in the one-key, group case.  ...  Among the considerations in [PRS02] , they showed that the 3-round Feistel cipher over abelian groups was not super pseudorandom, but left as an open problem a proof over non-abelian groups.  ... 
arXiv:1707.01699v2 fatcat:btl7ikrsqbcsjotgrj3xqjitpy

Page 7178 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
Summary: “New families of biphase sequences of size 2’~' +1, r being a positive integer, are derived from families of interleaved maximal-length sequences over Z, of period 2(2’ — 1).  ...  Vanstone have proposed a public- key cryptosystem (FGM) which is based on factorizations of a binary vector space (i.e., transversal logarithmic signatures of an elementary abelian 2-group).  ... 

Large connected strongly regular graphs are Hamiltonian [article]

László Pyber
2014 arXiv   pre-print
We prove this by showing that, apart from three families, connected strongly regular graphs are (highly) pseudo-random. Our results suggest a number of new questions and conjectures.  ...  Assume that the groups X n = Aut(G n ) are primitive permutation groups and that the index of the largest abelian normal subgroup of X n goes to infinity.  ...  Let P be a permutation group of degree n. If V is an n dimensional vectorspace over the complex numbers then P acts on V in a natural way by permuting the elements of an orthonormal basis.  ... 
arXiv:1409.3041v1 fatcat:mwvli7nk7jdlzm4lml2n54lxpm

Near-Optimal Cayley Expanders for Abelian Groups [article]

Akhil Jalan, Dana Moshkovitz
2021 arXiv   pre-print
We give an efficient deterministic algorithm that outputs an expanding generating set for any finite abelian group.  ...  Our technique is an extension of the bias amplification technique of Ta-Shma (2017), who used random walks on expanders to obtain expanding generating sets over the additive group of n-bit strings.  ...  our study of ǫ-biased sets over arbitrary abelian groups.  ... 
arXiv:2105.01149v1 fatcat:o3qjth3rbvbt5po3eg4bvoo63y

On the Lai-Massey Scheme [chapter]

Serge Vaudenay
1999 Lecture Notes in Computer Science  
We also show that this design offers nice decorrelation properties, and we propose a block cipher family called Walnut.  ...  Constructing a block cipher requires to define a random permutation, which is usually performed by the Feistel scheme and its variants.  ...  On Super-Pseudorandomness 6 A New Family of Block Ciphers In this section we construct a new family of block ciphers called Walnut (as for "Wonderful Algorithm with Light N-Universal Transformation") Walnut  ... 
doi:10.1007/978-3-540-48000-6_2 fatcat:dqdktmr4orc5zhbcqdt7mp6aji

The q-ary image of some q/sup m/-ary cyclic codes: permutation group and soft-decision decoding

J. Lacan, E. Delpeyroux
2002 IEEE Transactions on Information Theory  
Index Terms-Permutation groups, -ary image of -ary cyclic codes, soft-decision decoding.  ...  Using a particular construction of generator matrices of the -ary image of -ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation groups.  ...  Some infinite families of RS codes also satisfy this property.  ... 
doi:10.1109/tit.2002.1013146 fatcat:iew3vhmygfg2volkqlaamcloa4

More on additive triples of bijections [article]

Sean Eberhard
2017 arXiv   pre-print
We study additive properties of the set S of bijections (or permutations) {1,...,n}→ G, thought of as a subset of G^n, where G is an arbitrary abelian group of order n.  ...  Let G be an abelian group of order n.  ...  Similarly, a pseudorandom permutation is a family of permutation f s : {0, 1} d → {0, 1} d such that • for any s and x ∈ {0, 1} d there is an efficient algorithm for computing f s (x), • if s is chosen  ... 
arXiv:1704.02407v1 fatcat:wedyhxf5tnapdam3hxdncdqrna

Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus

Harald Niederreiter, Igor Shparlinski
2000 Acta Arithmetica  
We write U m = (Z/p m Z) * for the group of reduced residue classes modulo p m , where we drop the dependence on p in the notation for simplicity (we may think of p as a fixed prime).  ...  Here and in the following, it will often be convenient to write u/v for an expression uv −1 in a multiplicative abelian group. Lemma 1.  ...  The exponential sums (3) are relevant in the analysis of a well-known family of pseudorandom numbers.  ... 
doi:10.4064/aa-92-1-89-98 fatcat:bnu36y4qsjds3ghrro4l7ueasy

Pseudorandom Functions: Three Decades Later [chapter]

Andrej Bogdanov, Alon Rosen
2017 Tutorials on the Foundations of Cryptography  
Pseudorandom permutations (PRPs, see Section 5.2), which are closely related to PRFs, model block ciphers such as DES and AES, where the PRP security notion was a criterion in the design [117] .  ...  We say that games H 0 , H 1 are (t, ε)-indistinguishable if for every oracle circuit D of size at most t, where the probabilities are taken over the coin tosses of H 0 , H 1 .  ...  Let S and G be Abelian groups.  ... 
doi:10.1007/978-3-319-57048-8_3 fatcat:dwdqcxanardkthw4oon7qn7aia
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