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By Raphael Pass, Wei-lung Tseng

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What went wrong? The base case and the inductive step is perfectly valid! There are many “solutions” to this paradox, one of which is to blame it on the vagueness of the word “heap”; the notion of vagueness is itself a topic of interest in philosophy. Induction and Rationality: the Traveller’s Dilemma Two travelers, Alice and Bob, fly with identical vases; the vases get broken. The airline company offers to reimburse Alice and Bob in the following way. Alice and Bob, separately, is asked to quote the value of the vase at between 2 to 100 dollars.

1 Divisibility A fundamental relation between two numbers is whether or not one divides another. 1 (Divisibility). Let a, b ∈ Z with a = 0. We say that a divides b, denoted by a|b, if there exists some k ∈ Z such that b = ak. 2. 3|9, 5|10, but 3 7. The following theorem lists a few well-known properties of divisibility. 3. Let a, b, c ∈ Z. 1. If a|b and a|c then a|(b + c) 2. If a|b then a|bc 3. , transitivity). 37 38 number theory Proof. We show only item 1; the other proofs are similar (HW). By definition, a|b ⇒ there exist k1 ∈ Z such that b = k1 a a|c ⇒ there exist k2 ∈ Z such that c = k2 a Therefore b + c = k1 a + k2 a = (k1 + k2 )a, so a|(b + c).

Certainly we do not want to increase the size of N . If we apply the same solution as we did for encryption — break the message into chunks and sign each chunk individually — then we run into another security hole. Suppose Alice signed the sentences “I love you, Bob” and “I hate freezing rain” by signing the individual words; then Eve can collect and rearrange these signatures to produce a signed copy of “I hate you, Bob”. The solution again relies on the crazy hash function H: we require H to accept arbitrary large messages as input, and still output a hash < N .

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