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Is there a formal definition for a protocol that expands keys that are pre-shared among 2 or more parties?
How about an informal definition?

Example:

Alice and Bob pre-share a $n$-bit string $k$. After running algorithm $\Pi$ Alice and Bob have a common ($n$+$q$)-bit string $k'$, where $q>0$. In addition Eve learns nothing about $k$ or $k'$. Eve is an an eavesdropper on the channel between Alice and Bob with unlimited computational power. i.e. If $P_x$ are the odds of Eve learning $x$ then $P_k\le \frac{1}{2^n}$ and $P_{k'}\le \frac{1}{2^{n}}$.

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  • $\begingroup$ If Eve learns $k$, then she automatically learns $k'$ through $\Pi$, but your formulae seem to imply that $P_{k'} < P_k$. Are you perhaps looking for [en.wikipedia.org/wiki/Key_schedule](a key schedule), or rather [en.wikipedia.org/wiki/Key_stretching](key stretching)? $\endgroup$ – Ruben De Smet Jun 11 '17 at 15:12
  • $\begingroup$ @RubenDeSmet: That is true. I changed the bound for $P_{k'}$ I was looking at the wikipedia pages but none gave a definition of what it meant for a key expansion algorithm to be secure. For example would Alice and Bob seeding an agreed upon PRNG with $k$ and retrieving $n+q$ bits constitute as an information-theoretically secure key expansion? $\endgroup$ – edggy Jun 11 '17 at 15:33
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    $\begingroup$ This sounds more like a KBKDF (key based key derivation function) to me than key stretching or key schedule. Basically you team the input secret with a label and output size and then generate a key that is derived from the input key material $k$. An example is HKDF (with somebody posting the SHAKE equivalent in a minute or so :) ) $\endgroup$ – Maarten Bodewes Jun 11 '17 at 15:35
  • $\begingroup$ @MaartenBodewes: There are plenty of examples, but I can't find a definition. $\endgroup$ – edggy Jun 11 '17 at 15:51
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NIST SP-800 108 seems to have a semi-formal definition in chapter 5.0:

Section 1:

A key derivation function iterates a pseudorandom function (ed: PRF) $n$ times and concatenates the outputs until $L$ bits of keying material are generated, where $n = ⎡L/h⎤$.

Section 2:

For each of the iterations of the PRF, the key derivation key $K_I$ is used as the key, and the input data consists of an iteration variable and a string of fixed input data.

But I must admit that this seems more a description of the inner workings of the described KDF's than a formal definition of one, if such a thing exists.

As about any KDF actually works this way it remains to be seen if this isn't enough of a formal description.


The original HKDF paper has this more formal definition:

Definition 5: A key derivation function (KDF) accepts as input four arguments:

  • a value $\sigma$ sampled from a source of keying material (Def. 6),
  • a length value $l$,

and two additional arguments,

  • a salt value $r$ defined over a set of possible salt values
  • and a context variable $c$,

both of which are optional, i.e., can be set to the null string or to a constant. The KDF output is a string of $l$ bits.

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  • $\begingroup$ The original HKDF paper also should have plenty of definitions. eprint.iacr.org/2010/264 $\endgroup$ – SEJPM Jun 11 '17 at 16:02
  • $\begingroup$ I don't think this is what I'm looking for. I am looking for a multi-party (2) protocol in which a pre-shared is elongated securely. So far this looks like a one-party protocol. There is no mention of an eavesdropper or communication. $\endgroup$ – edggy Jun 11 '17 at 16:08
  • $\begingroup$ You mean something like a ratchet? I think there are several ways of doing so, so that may mean more than one definition I suppose... One thing you can do is to pick a KDF and then include a counter in the label. As long as the counters are kept in sync this can create any amount of key material. $\endgroup$ – Maarten Bodewes Jun 11 '17 at 16:12
  • $\begingroup$ As long as the key derivation is deterministic (which it is), you can run it at both sides and get the same extended key. That's probably not what you're after though. $\endgroup$ – Ruben De Smet Jun 11 '17 at 16:14
  • $\begingroup$ @edggy What you want is either a KBKDF as Maarten has it explained here or a pre-shared key authenticated key exchange (where you derive new keys using your old symmetric ones). Expansion of key material as a multi-party protocol doesn't make much practical sense and is usually actually a composition of the above named primitives. $\endgroup$ – SEJPM Jun 11 '17 at 16:14

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