## Wednesday, September 19, 2007

### Y'arr! Cryptography and piracy.

Ahoy, me mathematical mateys. Seein' as how today be the International Talk Like a Pirate Day, I decided to make a piratey post. Y'arr.

Y'arr! Here be a rough formal definition o' what ye can use for public-key encryption. Arr. A function f:Zx(Z/nZ) -> Z/nZ be a trapdoor function if:

1. Ye may compute f(m,k) smartly (in polynomial time).
2. The problem o' findin' a preimage o' k \in Z/nZ, given fixed m, is in NP.
3. There be some t not dependin' on k so, given t, findin' a preimage o' k with fixed m is in P.

Aye, I claim that these be the functions from which ye can construct a public-key cryptosystem. If ye disagree with me, ye can walk the plank. Otherwise, we'll continue:

Call the complexity class which consists o' findin' preimages under trapdoor functions PK (for public-key. Or plank, which be like a trapdoor, but better suited for seafarin' folk.). Is PK contained in FBQP?

If the answer be "aye," then pirates rejoice! With a quantum computer, ye can crack any classical public-key system, which would allow ye to commandeer credit card information and much, much more, without the risks of scurvy and losin' limbs that make pirates' health-insurance premiums so #@#\$in' high.

However, I believe that the answer be "nay." Since P is in BQP, this would imply P < NP, so I'm not expectin' any o' ye landlubbers to solve it. But if ye can give me evidence of why my hunch is correct, this ol' scurvy dog'll be forever indebted. Y'arr.

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