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Let's pretend that all digits of pi are known and arbitrarily long sequences of digits are trivial to get. Further, some mathematician proves that there are no patterns in pi. We could create a stream cipher by grabbing a piece of pi as long as our plaintext and combining the two with some function (such as XOR or modulus addition.) The key would be the starting position in pi.

Would this be equivalent (in terms of security) to a one-time pad? To what sort of attacks would it be vulnerable?

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The problem with this approach is that it literally gains you nothing. In order to choose a random subsequence of a needed length from $\pi$, you need to generate a cryptographically random number of at least the same length of the desired key to use as the offset. But then you may as well just use that number as your secret key.

Other than that, yes, it's exactly the same as a one-time pad. Just with a silly and pointless key derivation protocol which cannot mathematically increase the security of the system, but could conceivably weaken it.

Edit: As Thomas points out in the comments, the distribution of digits of $\pi$ are not random, and so this mechanism of key generation actually discards a significant amount of entropy that had been generated while choosing a random offset.

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The $n$th digit of $\pi$ is most certainly not a uniform random variable over $\mathbb{Z}_{10}$, therefore claiming that this scheme is equivalent to an OTP is incorrect (it may seem that choosing a random "starting position" is enough, but it's not - the underlying distribution of $\pi$ actually matters and if it is not uniformly random, it will destroy entropy encoded in said starting position). – Thomas Jan 26 at 5:30
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One of the original stated assumptions was "there are no patterns in $\pi$", from which I (perhaps incorrectly) assumed he meant something along the lines of "the distribution of digits in $\pi$ is random". Certainly if the the digits of $\pi$ are not random (as you've correctly stated is the case), this mechanism is significantly weaker than a true OTP. – Stephen Touset Jan 26 at 5:48
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Ah, I did not see Joshua's assumption - sorry! That'll teach me to comment without reading the entire thread.. – Thomas Jan 26 at 5:52
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Nothing we've not all been guilty of at some point. Regardless, it was absolutely worth pointing out. – Stephen Touset Jan 26 at 5:57
@Thomas: I know not result hinting at any bias in the digits of $π$ in any base; or at a distinguisher that would, in time independent of $n$, recognize with sizable advantage a fixed-length uniform random sequence from an equal length extract of the digits of $π$ starting at the $n$th. Is there such a thing? – fgrieu Jan 26 at 14:08
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