# The security of many round Luby Rackoff pseudo random permutations

Maurer U, Pietrzak KZ. 2003. The security of many round Luby Rackoff pseudo random permutations. EUROCRYPT: Theory and Applications of Cryptographic Techniques, LNCS, vol. 2656, 544–561.

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Author

Maurer, Ueli M;
Pietrzak, Krzysztof Z

^{ISTA}^{}Series Title

LNCS

Abstract

Luby and Rackoff showed how to construct a (super-)pseudo-random permutation {0,1}2n→ {0,1}2n from some number r of pseudo-random functions {0,1}n → {0,1}n. Their construction, motivated by DES, consists of a cascade of r Feistel permutations. A Feistel permutation 1for a pseudo-random function f is defined as (L, R) → (R,L ⊕ f (R)), where L and R are the left and right part of the input and ⊕ denotes bitwise XOR or, in this paper, any other group operation on {0,1}n. The only non-trivial step of the security proof consists of proving that the cascade of r Feistel permutations with independent uniform random functions {0,1}n → {0,1}n, denoted Ψ2nr is indistinguishable from a uniform random permutation {0,1}2n → {0,1}2n by any computationally unbounded adaptive distinguisher making at most O(2cn) combined chosen plaintext/ciphertext queries for any c < α, where a is a security parameter. Luby and Rackoff proved α = 1/2 for r = 4. A natural problem, proposed by Pieprzyk is to improve on α for larger r. The best known result, α = 3/4 for r = 6, is due to Patarin. In this paper we prove a = 1 -O(1/r), i.e., the trivial upper bound α = 1 can be approached. The proof uses some new techniques that can be of independent interest.

Publishing Year

Date Published

2003-06-04

Volume

2656

Page

544 - 561

Conference

EUROCRYPT: Theory and Applications of Cryptographic Techniques

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### Cite this

Maurer U, Pietrzak KZ. The security of many round Luby Rackoff pseudo random permutations. In: Vol 2656. Springer; 2003:544-561. doi:10.1007/3-540-39200-9_34

Maurer, U., & Pietrzak, K. Z. (2003). The security of many round Luby Rackoff pseudo random permutations (Vol. 2656, pp. 544–561). Presented at the EUROCRYPT: Theory and Applications of Cryptographic Techniques, Springer. https://doi.org/10.1007/3-540-39200-9_34

Maurer, Ueli, and Krzysztof Z Pietrzak. “The Security of Many Round Luby Rackoff Pseudo Random Permutations,” 2656:544–61. Springer, 2003. https://doi.org/10.1007/3-540-39200-9_34.

U. Maurer and K. Z. Pietrzak, “The security of many round Luby Rackoff pseudo random permutations,” presented at the EUROCRYPT: Theory and Applications of Cryptographic Techniques, 2003, vol. 2656, pp. 544–561.

Maurer, Ueli, and Krzysztof Z. Pietrzak.

*The Security of Many Round Luby Rackoff Pseudo Random Permutations*. Vol. 2656, Springer, 2003, pp. 544–61, doi:10.1007/3-540-39200-9_34.