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I have the need for a hexadecimal token that is smaller than the normal length of the hexadecimal representation of a SHA-256 hash.

Should I take the first bits or the last bits? Which of them contain the most entropy?

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If taking the first or last bits of a SHA-256 output made any difference, it would be viewed as a serious blow against the security of SHA-256. Right now, no such weakness is known in SHA-256. So, as far as we know, you can use whatever bits you want.

If you need a more "administrative" answer, have a look at SHA-224 (also specified in FIPS 180-3). This is a hash function with a 224-bit output, which NIST defined in order to "match" (administratively speaking) the proclaimed 112-bit security of 3DES. They defined SHA-224 by taking SHA-256, changing the conventional IV, and truncating the output. For the truncation, they took the first 224 bits. If it is good for NIST then it is good for you: take the first bits.

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    $\begingroup$ They only claimed that the first bits were strong right? crypto.stackexchange.com/questions/3153/… In fact, using the last bits would be detrimental to security no? $\endgroup$
    – Pacerier
    Commented Jul 9, 2012 at 19:33
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    $\begingroup$ @Pacerier - No. There is no reason to believe that using the last bits would be detrimental to security (and there are very strong reasons to expect that it is not detrimental). You're misreading the contents of that link. $\endgroup$
    – D.W.
    Commented Oct 23, 2013 at 6:44

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