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1 Constructing Pseudo-Random Permutations with a Prescribed Structure Moni Naor Weizmann Institute Omer Reingold AT&T Research

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2 Pseudo-Random Permutations Pseudo-random Permutations F 0,1 k 0,1 n 0,1 n key Domain Range F -1 0,1 k 0,1 n 0,1 n key Range Domain Family k F S S 0,1 k is pseudo-random if: –X F S -1 F S (X)) - Invertability –Succinct Representation: k log ( n !) –Efficiently computable: given S can compute F s and F S -1 –Indistinguishable from random permutations...

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3 Indistinguishability The tester T that can choose adaptively –X 1 and get Y 1 F S (X 1 ) –Y 2 and get X 2 F S -1 (Y 2 ) – –X q and get Y F S (X q ) Challenge: T has to decide whether F S R k or F S R (n) F F 0,1 n 0,1 n S

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4 (t, ,q)-pseudo-random For a function F chosen at random from (1) k F S S 0,1 k (2) (n) F F 0,1 n 0,1 n For all t-time machines T that get to choose q queries and try to distinguish (1) from (2) Pr T ‘1’ F R k - Pr T ‘1’ F R (n) Want a family where is negligible as long as t and q are not too large

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5 Model Block Ciphers Block-Ciphers : Shared-key encryption schemes where the encryption of every plaintext block is a ciphertext block of the same length. Important Examples: DES, Rijndael (AES) ey CC Plaintext Ciphertext

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6 Construction of Pseudo-Random Permutations Defined and constructed by Luby and Rackoff Possible to construct p.r. permutations from p.r. functions (and vice versa...) Based on 4 Feistal Permutations - 2 of which should be pseudo-random functions. f L1L1 R1R1 L2L2 R2R2

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7 Permutations with a Prescribed Structure Example: Cyclic Permutations Want to construct a family of permutations that is Pseudo-Random Cyclic Motivation: a never repeating, random looking sequence X 1, X 2,...,X i,... such that X i+1 =F S (X i ) [Shamir-Tsaban]

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8 Permutations with a prescribed Structure A cycle type - list of how many cycles there are of each size Want to construct a family of permutations where Each member has cycle type C Pseudo-Random : –Succinct Representation: k log ( n !) –Efficiently computable: given S can compute F s and F S -1 –Indistinguishable from random permutations with cycle type C

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9 The Construction To construct C a p.r. family of permutation with type C: Let k F S S 0,1 k be a family of pseudo- random permutations Let be a (fixed) permutation with cycle type C C P S F S F S -1 S 0,1 k To evaluate P S (X): compute F S -1 ( (F S (X))) To evaluate P S -1 (Y): compute F S -1 ( -1 (F S (Y)))

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10 The Construction... Example: cyclic permutation (X) X+1 mod n Complexity of evaluation: Two invocations of F S (one in each direction) One invocation of

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11 Why does it work? Well known theorem from elementary group theory: For any two permutations and and -1 have the same cycle type. Prove a stronger statement: Theorem 1 : For any permutation with cycle type C, let be a random permutation. Then the permutation -1 is uniformly distributed over the permutations with cycle type C.

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12 Security of Construction Theorem 2 : Suppose that adversary D can distinguish with advantage whether a given permutation is R C or a random permutation of type C. Then there is a D’ can distinguish the family k from (n) with advantage . Running time of D’ is t running time of D. t time to evaluate and -1

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13 Proof by Simulation D’ is given as a black-box. It simulates D on -1 –When D queries a point X - D’ requests X) and then -1 at point X –When D queries an inverse of point X - D’ requests X) and then -1 at point -1 X –Outputs the same guess as D From Theorem 1 the probabilities of distinguishing are identical.

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14 Involutions An involution is a permutation that is self -inverse When used for encryption - the encryption and decryption operations are identical. Let X X+1 if x is even and X X-1 if odd. Resulting I is a family of involutions with no fixed points.

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15 Combinatorial Randomness (almost) t-wise independence - combinatorial counterpart to (cryptographic) pseudo-randomness If instead of k a family H of 2t-wise independent permutations is used, the result is –a t-wise independent family of permutations with cycle type C. If an approximation to 2t-wise is used - similar approximation in c

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16 Fast Forward Possible to iterate P S c with ‘zero’ cost: P S (m) ( X )= F S -1 ( (F S (F S -1 ( )))= F S -1 ( (m) (F S (x))) Same as iterating In case of cyclic permutations: P S (m) ( X )= F S -1 (F S (x) +m mod n ) Also easy to check whether X 1 and X 2 are in the same cycle.

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17 Open Problems Fast forward property for permutations with no prescribed cycle type. –Sufficient to find right distribution on cycle types. Fast forward property for pseudo-random functions – Algorithmic applications: Pollard’s , Hellman time- space tradeoff – Caveat - does not necessarily improve them Construct pseudo-random permutation of size N’ < N given one of size N.

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18...Open Problems Other combinatorial structures - is it possible to generate a succinct/implicit representation that looks random of Pseudo-random graphs –G n,p or bounded degree –Involution - d regular d colorable Latin Squares –2 n 2 n matrix where each row and each column are a permutation of 0,1 n –Non trivial even for non-implicit

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