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The Hash Function “Fugue” Shai Halevi William E. Hall Charanjit S. Jutla IBM T. J. Watson Research Center

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Broad Overview Maintains Large State (~1000 bits) –Large Initial State Tougher to invert –Large Final State Tougher to find collision Final Compression (to say, 256 bits) –Lots of Crunching Tougher to find “properties”, invert, or find collision Uses ‘super’ AES-like rounds Focus of the Talk : Collision Resistance

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Initial State (30 words) Process New State M1 M_i Final Stage Hash Output (8 words) Iterate State Fugue-256

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Initial State (30 words) Process New State ΔM1 ΔM_i Final Stage Δ = 0 Iterate State Fugue-256 State Δ = 0

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Overview (contd.) Inspired by Grindahl [KRT07] –Small incremental input rounds –Long final stage. –Attacked by Peyrin [P07] -Internal Collisions Fugue has a Proof Driven Design –Proves that Peyrin-style attacks do not work Proves bound on differential attacks assuming limits (extremely generous) on message modification. –Proves bound on finding External Collisions. –Like AES, uses MDS codes but bigger MDS codes –Does not need MD mode theorem

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Design Challenges Non-secret Key/ Un-keyed properties gives adversary “non-standard” approaches to enhance differential attacks –Message Modification –Neutral Bits, Neutral Differentials

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Good News The main “non-secret key” properties are about collisions –Collision, TCR, 2 nd Pre-image, Universal Hash –The differential requires output difference to be zero, i.e. out = 0. –Is it easy to prove something about such restricted differentials? Quandary: All good practical designs are based on permutations.

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Initial State (30 words) Process New State ΔM1 ΔM_i Final Stage Δ = 0 Iterate State Δ 0 Fugue-256 Invoke Coding Theory

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What’s in an elementary round? [Called SMIX in the paper] Works on 128 bits (just like AES) Arranged as 4 by 4 matrix (just like AES) Starts with S-box substitution (same as AES) Does linear mixing (more advanced than AES)

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AES Round

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Leads to MDS code over 16 bytes! Fugue elementary round “SMIX”

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Leads to MDS code over 16 bytes! Fugue elementary round “SMIX”

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Leads to MDS code over 16 bytes! At least 13 active S-Boxes 2^{-6*13} = 2^{-78} Fugue elementary round “SMIX”

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FINAL STAGE Rapid Mixing (G1) Differential Killer (G2) output

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External Collision Provable Bound Assumption: Differential Attacks –Attacker controls difference, state itself is random –Probabilities of different rounds are assumed independent. Consider 2 messages leading to two different states at middle of final stage –After the rapid mixing Allow the adversary to force a difference D of its choice at this point.

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External Collision Provable Bound Theorem: For any state difference D 0, if the states at the start of G2 are chosen randomly then Pr[ Collision in 256 bit output | D ] 2 -129 Recall, assumes independence assumption

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Initial State (30 words) Process New State Final Stage Iterate State Fugue-256 Process M1 SMIX M1 Repeat once more

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Process State M, M’_i State _i = 0 Process State M, M’_i-1 Process State M, M’_i-2 Process Random States, but _{i-4} = D M, M’_i-3 Theorem Pr [ M, M’ : _i =0 | _{i-4} = D ] 2 -168 INTERNAL COLLISION PROVABLE BOUND

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Message Modification? Neutral Bits? How justified is random state? We do more advanced analysis, giving extremely generous “free message modification / all bits neutral ” allowance. –Still can prove 2^{-128} bound.

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Performance (Fugue-256) 32 bit Intel Core 2 Duo (Linux) –ANSI C : 36 cycles/byte 64 bit Intel Xeon (CygWin) –ANSI C : 28 cycles/byte 8-bit: as good as AES…similar advantages –Decent state size : 120 bytes (1000 cycles/byte) Hardware: 90nm IBM Cu-8 technology –360 MB/sec (basic) to 1.8 GB/sec.

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Conclusion Proof-driven Design leads to best of both worlds: - Security - Performance

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THE END

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Fugue is a Universal Hash Fn. Requirement: –For all messages M1,M2, Pr_k[ Fugue_k(M1) = Fugue_k(M2)] is low Key is 8 words (256 bits), placed in the right most 8 words of initial state. Assume (for now) same length. Wlog internal collision, otherwise messages irrelevant.

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Internal coll. at end of Round 0 Number the rounds backwards 0, -1, -2,… For now, assume states at start of round –3 are random (but say, with some adversary determined difference D). Then, we have already proven 2^{-168}.

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What about the random assump. Random but diff = D at start of round –3. –That is allowing adversary to get an XOR-diff D with probability 1 (on a random key)! –So seems not that bad an assumption. But, is entropy depletion a problem? –State starts with 8 word entropy. –Each round adversary inserts a word (pair).

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Input State (30 words) Process Output State (O_S) M1 SMIX D Repeat once more I_S M1 (D; O_S) => ! (M1;IS) (D1,O_S) ->(M1; IS) (D2,O_S) ->(M1; IS’) ? Nope!

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Input State (30 words) Output State (O_S) = 0 X SMIX D I_S M1 (D1,O_S) ->(M1; IS) (D2,O_S) ->(M1; IS’) ? M1 X SMIX 0 All 4 words non-zero X non-zero All 4 words non-zero M1 non-zero

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Discarded Slides

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Desired Properties Keyed Properties –Secret Key PRF (MAC) –Non-Secret Key (Salted) Universal Hash, Extractor, Key Derivation Collision Resistance, TCR, Pre-image (1 st / 2 nd ) Un-Keyed Versions Collision Resistance, Pre-image (1 st /2 nd )

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