By Richard Guaraldo, Susan M. Zaker

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Index i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 vector index index vector index index vector xi−1 i xi−1 i xi−1 e0000000 33 17 80000002 65 49 2 2 34 18 0 66 50 0 2 35 19 2 67 51 0 80000000 36 20 0 68 52 0 1 37 21 3 69 53 0 0 38 22 0 70 54 0 80000001 39 23 2 71 55 0 2 40 24 2 72 56 0 40000002 41 25 1 73 57 0 2 42 26 0 74 58 0 2 43 27 2 75 59 0 80000000 44 28 2 76 60 0 2 45 29 1 77 61 0 0 46 30 0 78 62 0 80000001 47 31 0 79 63 0 0 48 32 2 80 64 0 40000001 49 33 3 81 65 4 2 50 34 0 82 66 0 2 51 35 2 83 67 0 80000002 52 36 2 84 68 8 1 53 37 0 85 69 0 0 54 38 0 86 70 0 80000001 55 39 2 87 71 10 2 56 40 0 88 72 0 2 57 41 0 89 73 8 2 58 42 0 90 74 20 0 59 43 2 91 75 0 0 60 44 0 92 76 0 1 61 45 2 93 77 40 0 62 46 0 94 78 0 80000002 63 47 2 95 79 28 2 64 48 0 96 80 80 Finding Collisions in the Full SHA-1 31 Table 6.

U} of k indices and a guess of the values of {bi | i ∈ S}. Then the probability that all k guesses are correct is at most 2B /2k , where the probability is over the random variables and the coin ﬂips of the hint generation and the reconstruction procedures. Proof (of claim). Fix an arbitrary sequence of random choices for the reconstruction procedure. Each ﬁxed hint yields a choice of S and a guess of the bits of S. For any ﬁxed hint, the probability, over choice of b1 , . . , bu , that all the guessed Pebbling and Proofs of Work 45 values are consistent with the values of the elements of S is 2−k .

Bu ), produces a subset S ⊂ {1, . . u} of k indices and a guess of the values of {bi | i ∈ S}. Then the probability that all k guesses are correct is at most 2B /2k , where the probability is over the random variables and the coin ﬂips of the hint generation and the reconstruction procedures. Proof (of claim). Fix an arbitrary sequence of random choices for the reconstruction procedure. Each ﬁxed hint yields a choice of S and a guess of the bits of S. For any ﬁxed hint, the probability, over choice of b1 , .