imperfect info -> perfect security

imperfect info -> perfect security

1. Privacy amplification

   • Turns a weak secret into a strong one, by doing a process that sepeartes the density matrix of eve out into a tensor product.
   • Alice and Bob need to synchornise at some point as its important to ensure that alice and bob are able to communicate accuratley while still keeping eve out of the loop. However given the setup below, eve can see the initial string x, and can have some information to use to identify the message between alice and bob. So alice and bob need to randomise their string, but need to do so in a way that they both end up with the same randomised string R (Ra = Rb). So A and B must interact to allow this
     • For secrecy to be valid need the trace to be less than the secret parapmeter episioln s
     • Also need t assume there is limit on how secret you can be from the conditional entorpy
   • amplifying a weak secret by parity
     • Applied using the parity. As long as Eve doesn't know all of the bits chosen to be random by Y, even if there is only one, then the parity will be random

2. randomness extractors

   • Similar to privacy extraction but alice doesn't need to coordinate with a Bob
     • It is directly comparable to a previous example. The Privacy amplification from parity uses a random key string made by alice which she can publically send to bob
       • With same secrecy requirement
         • If min entroupy is at least K, then have that the density of the seed with eve is indistinguishable from the totally mixed state
       • Difference is that instead of just saying Alice makes up a random number string Y, we name y as a random extarctor, which would be a random number generator in alices lab, which forms y and then send this to Bob. Then A and B can find Z = ext(X,Y) = ext (X,Y) (= Ra = Rb) Y could also be called the seed
   • Definition
   • strong seeded extractor

3. 1-universal hash function. An extracter is 1-univerasl if for a fixed source X, if we apply a random seed y, then the output will be a uniformly distributed string.

   • Example
   • 2 - universal if true for 2 unique values of x
     • e.g the inner product is 2-univeral, however the bitwise addition (example used for 1-universal) isn't 2-universal
     • Collison probability must be bounded (limit on how uniform a 2-univerasl will be)
       • Can find a trace (statistical ) distance from this. Uselfully this shows us that the correctness parameter can be very good with 2-universal method. As long as we try to extract fewer bits (M) than there are unpredictable bits (K). This is written as a lemma called "The Leftover Hash Lemma"...
         • The 2-universal is slightly limited as the seed length is typically long, this isn't a major issue, and in the future ways will be shown to reduce the length

4. hash functions are generally functions thar convert a long string into a shorter string. In out uses of hash functions we also have the requirment that they lead to the shorter output string be well distributed.

5. pretty good measurement

   • Classical method with more than 2 measurement operators
   • with Quantum side info
   • cauchy shwartz inequality
   • example

6. hash means reduce the string length

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