Analysis of Non-fortuitous Predictive States of the RC4 Keystream Generator

Analysis of Non-fortuitous Predictive States of the RC4 Keystream Generator Souradyuti Paul and Bart Preneel K.U. Leuven, ESAT/COSIC Indocrypt 2003 India Habitat Center December 8, 2003

Overview of the Presentation • Description of RC4 • Definition of a Predictive State and its Importance • Upper Bound on the Number of Outputs of a Predictive State • Definition of a Non-fortuitous Predictive State • Determination of Non-fortuitous Predictive States • Of Length 1 and 2 • General Approach • Conclusions

Description of RC4 • Based on Exchange Shuffle Paradigm • The Algorithm Runs in Two Phases • Key-scheduling Algorithm • Pseudo-random Generation Algorithm • Pseudo-random Bytes are Bit-wise X-Ored with the Plaintext Bytes in Succession to Generate the Ciphertexts.

Key-scheduling Algorithm • A Variable Size Key (K) Turns an Array (S) of Identity Permutation into a ‘Random’ Permutation • The Size of the Key K=40to 256 Bits in All Practical Applications • The Size of the Array N = 256 Bytes in All Practical Applications

Key-scheduling Algorithm Input (S, K) 1. for (i = 0 to N-1) S[i] = i ; 2. j = 0; 3. for (i = 0 to N-1) j = (j + K[i mod l] + S[i] ) mod N; Swap (S[i], S[j] );

Pseudo-random Generation Algorithm Input (S) 1. i = 0; 2. j = 0; 3. i = i + 1; 4. j = (j + S[i] ) mod N; 5. Swap (S[i],S[j]); 6. I = (S[i] + S[j]) mod N ; 7. Output = S[I];

Pseudo-random Generation Algorithm Input (S) 1. i = 0; 2. j = 0; 3. i = i + 1; round 4. j = (j + S[i] ) mod N; 5. Swap (S[i], S[j]); 6. I = (S[i] + S[j]) mod N ; 7. Output = S[I];

Overview of the Presentation • Description of RC4 • Definition of a Predictive State • Definition of a Non-fortuitous Predictive State • Main Contributions • Upper Bound on the Number of Outputs of a Predictive State and its Importance • Determination of Non-fortuitous Predictive States • Of Length 1 and 2 • General Approach • Conclusions

Predictive States of RC4 • An a-state of RC4 is only a known elements of the S-box together with i and j at some round denoted by round 0. • In the next c rounds b output bytes are produced where c <= N where b>1 and round 1 produces output. • This internal state of RC4 at round 0 is defined to be b-predictive a-state.

Predictive States of RC4Snapshot at Round 0 Round: 0 1 … … r …. c i j Number of Known elements in the S-box is a.

Predictive States of RC4Snapshot at Roundc Round: 0 1 … … r …. c i j Outputs:Z1 Z2 Z3 …… Zb Number of Predicted Outputs is b.

Non-fortuitous Predictive States • Consider the a-predictive a-states. • If a elements of the S-box are consecutive and so are the a outputs then the state is a Fortuitous State of length a. • All other a-predictive a-states are Non-fortuitous Predictive States of length a.

Main Contributions • We give an upper bound on the number of predicted outputs b for a b-predictive a-state • We also give an algorithm which is better than exhaustive search to determine Non-fortuitous Predictive States for small values of a

For a b-Predictive a-Stateb <= a (Sketch of the Proof) • The claim was left as a conjecture by Mantin and Shamir, 2001. • The bound on c, which was 2N in the original conjecture, is wrong. When a=N, b is infinitely large. • The claim is true when c <= N. • Clearly a-predictive a-states are important. • The proof is by contradiction.

For a b-Predictive a-Stateb <= a (Sketch of the Proof) • Assume b>a. • S[i] is always occupied with a known element at each round till the cth round is reached otherwise the execution is stopped. • Maximum one element can be filled in a vacant place in one round. • Maximum of (c-b) locations can be filled with known elements in c rounds. • Therefore, b known elements at round 0 leads to contradiction.

Importance of Predictive States when b = a • Assume Internal States and External States (i.e., Outputs) of RC4 are ‘random’ for a fixed i. • For Predictive States when b = a, the elements of the S-box elements can be predicted with the maximum probability, that is 1/N, when outputs are known. • The larger the number of a-predictive a-states the higher is the probability for one of them to occur.

Determination of Non-fortuitous Predictive States • An efficient algorithm to determine the Fortuitous States of small length is designed by Fluhrer and McGrew, 2000. • The main problems to determine the Non-fortuitous Predictive States are • The inter-element-gaps of the S-box elements are not known. • The inter-element-gaps of the S-box elements change after each round.

The Set of Non-fortuitous Predictive States of length 1 is Empty Index: 0 1 2 2x-1 2 x ... x N-1 i j • Any 1-predictive 1-state is a Fortuitous State. • The number of 1-predictive 1-states is N.

The Set of Non-fortuitous Predictive States of length 2 is Empty Index: 0 1 2 r … r’ N-1 i Outputs: Z1 Empty Index: 0 1 2 r … r’ N-1 i • Therefore, r’-r = 1, otherwise RC4 halts.

The Set of Non-fortuitous Predictive States of length 2 is Empty Index: p1 p2 p3 p4 p5 1 i j Outputs: Z1 • Finney’s Forbidden State after the 1st round. Therefore, not possible. Possibility 1

The Set of Non-fortuitous Predictive States of length 2 is Empty Index: p1 p2 p3 p4 p5 2 i j Outputs: Z1 After the 1st round Possibility 2

The Set of Non-fortuitous Predictive States of length 2 is Empty Index: p1 p2 p3 p4 p5 2 i j After the 2nd round Possibility 2

The Set of Non-fortuitous Predictive States of length 2 is Empty Empty Index: p1 p2 p3 p4 p5 2 i j After the 3rd round Possibility 2

Determination of Non-fortuitous Predictive States: A General Approach • The inter-element-gap is the number of vacant places between two successive elements of the S-box. • The possible inter-element-gaps of the a-predictive a-states are determined from that of (a-1)-predictive (a-1)-states recursively. • Once the inter-element-gaps are known then we apply an algorithm similar to the one by Fluhrer and McGrew, 2000.

Conclusions • We obtained an important combinatorial result that an a-state of RC4 can not produce more than a outputs in the next N rounds. • A practical algorithm is designed to determine a special set of RC4 states known as Non-fortuitous States which reduce the data complexity of all known attacks on RC4.

Analysis of Non-fortuitous Predictive States of the RC4 Keystream Generator