If you didn’t get a “gold star” on Checkup 2, you can improve your score by answering one or more of the questions below. You can submit your revisions in class Wednesday, or at the end of office hours Wednesday. To submit revisions, attach your answers to these questions to your original submission.

R1. Hashing (Answer this question if you want to improve your score on questions 1-4)

There are several different places in bitcoin where cryptographic hashes are used:

A. Producing the public bitcoin address by hashing the public key.

B. Producing a transaction digest for use as the input in signing a transaction.

C. Producing the Merkle tree root for authenticating the transactions in a block (using hashes all the way up the tree).

D. Producing the hash of the previous block to use in the block header.

E. Producing the double hash of the block (with nonces) to find a block that satisfies the difficult needed in mining.

Suppose H is a hash function that provides strong pre-image resistance, but does not provide collision resistance. That is, an adversary knows way to efficiently find pairs of values, m1 and m2 such that m1 is not equal to m2 and H(m1) = H(m2).

Explain clearly one place where cryptographic hashing is used in bitcoin that the adversary could gain an advantage by knowing how to find collisions in H. (Be careful to not overstate the advantage, though.)

R2. Blockchain (Answer this question if you want to improve your score on questions 5-7)

Ishtosa suggests strengthening bitcoin by adding the hash of the previous previous block to the block header. So, the block header for block 534 would now include the both the hash of the block 533, and the hash of block 532. Would this be a good idea?

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