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Suppose there is a cryptographic hash function $H : \{0,1\}^* \to \mathbb G$. Now, we want to define a hash function $H' : \{0,1\}^* \to \mathbb G \times \mathbb G $.

My first idea is to define $H'$ in the following way:

$H'(x) = (H(x||'0'), H(x||'1'))$

Is $H'$ cryptographically secure? If not, can you propose a secure construction that uses $H$?

The motivation behind this question is that some crypto libraries (e.g., Charm) provide functions for hashing to groups (e.g., $H$), but in some cases you need custom hash functions (e.g., $H'$)

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  • $\begingroup$ Two comments: 1) your construction looks very much like KDF2, which is probably secure. 2) you won't double your security using this construction (but I don't think you intended to do that). If $x$ is relatively small you could use any KDF, using $x$ as secret. $\endgroup$ – Maarten Bodewes♦ Apr 11 '16 at 12:22
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    $\begingroup$ You might want to clarify what it needs to be secure against. $\endgroup$ – Maarten Bodewes♦ Apr 11 '16 at 12:25
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    $\begingroup$ @MaartenBodewes "your construction looks very much like KDF2, which is probably secure". Do you mean "probably" or "provably"? Regarding the intended security notion, I wasn't targeting anything in particular... I guess the usual stuff (pre-image resistance, collision resistance, etc). I basically wanted to know if my original idea has some inherent problem, or if maybe there is a standard way to do this. $\endgroup$ – cygnusv Apr 11 '16 at 13:17
  • $\begingroup$ No, probably secure. Note that I asked the question about the security KDF1/2 constructs. Lookfor something based on HMAC such as HKDF for better proofs. $\endgroup$ – Maarten Bodewes♦ Apr 11 '16 at 13:21

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