Paper 2010/384

Practical consequences of the aberration of narrow-pipe hash designs from ideal random functions

Danilo Gligoroski and Vlastimil Klima

Abstract

In a recent note to the NIST hash-forum list, the following observation was presented: narrow-pipe hash functions differ significantly from ideal random functions $H:\{0,1\}^{N} \rightarrow \{0,1\}^n$ that map bit strings from a big domain where $N=n+m,\ m\geq n$ ($n=256$ or $n=512$). Namely, for an ideal random function with a big domain space $\{0,1\}^{N}$ and a finite co-domain space $Y=\{0,1\}^n$, for every element $y \in Y$, the probability $Pr\{H^{-1}(y) = \varnothing\} \approx e^{-2^{m}} \approx 0$ where $H^{-1}(y) \subseteq \{0,1\}^{N}$ and $H^{-1}(y) = \{x \ |\ H(x)=y \}$ (in words - the probability that elements of $Y$ are ``unreachable'' is negligible). However, for the narrow-pipe hash functions, for certain values of $N$ (the values that are causing the last padded block that is processed by the compression function of these functions to have no message bits), there exists a huge non-empty subset $Y_\varnothing \subseteq Y$ with a volume $|Y_\varnothing|\approx e^{-1}|Y|\approx 0.36 |Y|$ for which it is true that for every $y \in Y_\varnothing,\ H^{-1}(y) = \varnothing$. In this paper we extend the same finding to SHA-2 and show consequences of this abberation when narrow-pipe hash functions are employed in HMAC and in two widely used protocols: 1. The pseudo-random function defined in SSL/TLS 1.2 and 2. The Password-based Key Derivation Function No.1, i.e. PBKDF1.

Note: A typo is corrected in Lemma 3 (thanks to Ralph Wernsdorf from Rohde & Schwarz SIT GmbH)