Nostradarnus Attack

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Finally, we describe an interesting attack due to Kelsey and Kohno [80]that is applicable to any hash function that employs the Merkle-Damggrd construction (see Figure 5.1, above). The hash functions MD4 and MD5 discussed later in this chapter are of this type, as are the popular SHA-1 and Tiger hashes. Hash functions are often used in practice to prove prior knowledge or to commit to something without revealing the something. For example, suppose that Alice, Bob and Charlie want to place sealed bids online. Since they do riot trust that their bids will remain secret, neither Alice, Bob nor Charlie wants to submit their bid until the other two have submitted theirs. One possible solution to this problem is the following. First, Alice determines her bid A, Bob determines his bid B and Charlie determines his bid C. Then Alice submits h ( A ) ,Bob submits h ( B ) ,and Charlie submits h ( C ) . Once all three bids have been received, they are posted online, at which point Alice, Bob and Charlie can submit their respective bids, namely, A, B , and C . If the hash function h is one-way, there should be no disadvantage to submitting a bid before the other bidders. Also, if h is collision resistant, it should not be possible for Alice to change her bid once B and C have been revealed (and similarly for Bob and Charlie).4 Now consider the following scenario [80]. Trudy claims that she can predict the future. To prove it, on January 1, 2008 she publishes y, which she claims is h(z), where 2 gives the final Standard and Poor s 500 (S&P 500) index5 for December 31, 2008, along with various other predictions about events

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However, without modification, this protocol is insecure; see Problem 6. The S&P 500 is a well-known stock market index.

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HASH FUNCTIONS

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that will occur in 2009 and beyond. Suppose that on January 1, 2009, Trudy reveals a message z such that y = h ( z )and z correctly predicts the S&P 500 index for December 31, 2008, followed by a rambling set of predictions about future events that have not yet occurred. Does this prove that Trudy can foretell the future It would seem that by publishing y in advance, Trudy is committed to a specific x, unless she can violate the one-way property of the hash function h. It is generally much more difficult to violate the one-way-ness of a hash function than to find collisions. Barring any shortcut attack, if h generates an n-bit hash, then about 2n hashes need to be computed before Trudy could expect to find a message x that hashes to a specified value y, while only about 2n/2 hashes need to be computed to find a collision. So, in the scenario outlined in the previous paragraph, it would seem that if n is sufficiently large so that it is infeasible for Trudy to compute 2n hashes and no shortcut attack on h exists, then Trudy can legitimately clairn to be thc new Nostradamus. However, in [SO] it is shown that, Trudy can cheat, provided the hash function h uses the Merkle-Damggrd construction and that Trudy is able to compute collisions. By the birthday problem, if 2n/2 is a feasible amount of work (where the hash h generates an n-bit output), Trudy can find collisions. Of course, if there is a shortcut collision attack on h, then Trudy may be able to compute collisions even more efficiently than via the birthday attack. But for the remainder of this discussion, we assume that Trudy computes collisions by the birthday attack and that 2 12 is a feasible amount of work, while 2n is not. Under these assumptions, we describe how Trudy can cheat. Trudy must specify the value y in advance. Then when she knows the S&P 500 index for December 31, 2008, she wants to create a message that includes the S&P 500 result and hashes to y. More precisely, on January 1, 2009 Trudy knows the final S&P 500 index for December 31, 2008, and she sets P , the prefix, equal to this index result. Then Trudy must determine a suffix S so that h(P,S) = y, where y is the previously specified hash value. That is, the prefix P and y are specified, but Trudy is free to choose the suffix S so that y = h(P,S). Of course, if Trudy can randomly 2n suffixes S arid compute the corresponding hashes, then she would expect to find one for which y = h(P,S ) . But we assume that this is an infeasible amount of work, that is, Trudy cannot perform a brute force pre-image attack. Kelsey and Kohno [80] describe their attack as a herding attack, since a specified prefix P is herded into the specified hash value by selecting suffixes S and computing collisions (as opposed to pre-images). Furthermore, in this attack Trudy can has considerable control over S , so that it is possible for her to construct meaningful suffixes-instead of random gibberish, which

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Nostradamus (1503 ~1566)published numerous prophecies. His modern supporters clairn that he predicted most of the major events of recent history. Ironically, Nostradamus predictive powers seen1 to work best in retrospect.

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