GenericAttacksMerkleDamgaard
On this page, we describe attacks that apply on hash function that are designed according to the Merkle-Damgaard principle. Attacks that are even more generic and apply on all hash functions are described on this page
1 Collision style attacks
In case a hash function is not considered as a black box, but built from compression functions (which in turn are considered as black boxes at this point), multi-collisions can be constructed more efficiently. Ideally, the effort to find 2t single collisions should grow according to the birthday paradox: for t much smaller than n/2 the effort should grow almost linearly with each additional collision. What Joux showed in 2004 is that for iterated constructions the effort to find a 2t-multicollision is actually t*2n/2 The idea is to simply concatenate t collisions found by a birthday attack (or by any other mean like shortcut attacks for that matter). Since each collision allows to pick a message out of a pair of messages, and this choice is available t times, a set of 2t different messages consisting of t message blocks can be constructed that all lead to the same hash value.
An application of Joux's multicollisions is
the analysis of concatenated constructions. Assuming two hash
functions of output size n each whose outputs is concatenated, one
would ideally expect a security of 2n against birthday based
collision search attacks. Generating a 2n/2 multicollision for
one of the hash functions is however enough to find a collision in
the concatenated construction. This has a total cost of
2^{n/2+log(n)}.
As a historic note, it should be mentioned that Coppersmith's attack in 1985 on the Davies-Price variant of Rabin's scheme builds already on such a multicollision idea.
Antoine Joux - Multicollisions in Iterated Hash Functions. Application to Cascaded Constructions
- CRYPTO 3152:306-316,2004
- http://springerlink.metapress.com/openurl.asp?genre=article{\&}issn=0302-9743{\&}volume=3152{\&}spage=306
BibtexAuthor : Antoine Joux
Title : Multicollisions in Iterated Hash Functions. Application to Cascaded Constructions
In : CRYPTO -
Address :
Date : 2004
Don Coppersmith - Another Birthday Attack
- CRYPTO 218:14-17,1985
- http://link.springer.de/link/service/series/0558/bibs/0218/02180014.htm
BibtexAuthor : Don Coppersmith
Title : Another Birthday Attack
In : CRYPTO -
Address :
Date : 1985
2 Second preimage attacks
Discoveries about second preimage attacks on iterated hash functions span the last three decades. Merkle notes in 1979 that for messages of length 2k, the same number of different target hash values will speed-up the search for second preimages (of potentially different length) to 2n-k trials. See also extensions of Winternitz, Lai and Massey, and Preneel.
One of the reasons to include the message length as part of the
message to be hashed in constructions since then, is to prevent
these type of attacks. However, Dean describes in 1999 a way
to circumvent this measure by used so-called expandable messages.
Expandable messages are a set of messages of different lengths that
all yield the same intermediate hash value.
Dean's construction only works for compression functions that have easily constructed fixed-points, i.e. where it is easy to find a message block and an input chaining value that results into the same output chaining value. Many popular hash function construction indeed do have this property. In 2005, Kelsey and Schneier managed to remove this constraint and gave an algorithm to construct expandable messages for any compression function with an $n$-bit intermediate value. Their idea is to construct multicollisions out of collisions between message blocks of different length. From that, again example messages can be constructed and hence the search for second preimages is again of order 2n-k+1word.
John Kelsey, Bruce Schneier - Second Preimages on n-Bit Hash Functions for Much Less than 2^n Work
- EUROCRYPT 3494:474-490,2005
- http://dx.doi.org/10.1007/11426639_28
BibtexAuthor : John Kelsey, Bruce Schneier
Title : Second Preimages on n-Bit Hash Functions for Much Less than 2^n Work
In : EUROCRYPT -
Address :
Date : 2005
3 Preimage attacks
Herding attacks are a special kind of preimage attack, in the sense that an additional assumption is being made for the attack to work. The basic scenario in which herding attacks are applicable is as follows. At the cost of a pre-computation step, an attacker can commit to a digest of a hash function without yet knowing the input. In \cite{Kelsey2005HerdingHashFunctionsa}, this attack is described and shows that for all iterated hash functions the complexity is less than one would expect from an ideal hash function.
\begin{definition}[Resistance against herding attacks] Given a hash function h, the attacker may choose a digest H. If she is given P, it should not be possible to find S such that h(P||S)=H is considerably faster than by 2n- invocations of h. \end{definition}
For short suffixes, the workfactor for a herding attack on an iterative hash functions as shown in \cite{Kelsey2005HerdingHashFunctionsa} is $2^{(2n-5)/3}$. First a so-called diamond structure is built in a precomputation phase that results in a particular digest $H$. After $P$ is given to the attacker, a linking message $S_1$ is searched that connects $P$ with one of the edges of the diamond structure. Let's denote the path between the found entry point in the diamond structure and the digest $H$ at its end $S_2$, then the result string $S$ such that $h(P||S)=H$ is $S=S_1 || S_2$.
Besides observing this theoretical weakness, we can also consider
the feasibility in practice of this attack. In the case of SHA-1,
and without partial knowledge of $P$, a pre-computation effort of
$2^{107}$ would be needed to compute $H$. This requires about
$2^{60}$ bits of storage for the required data-structure.
Afterwards, $2^{107}$ effort would be needed to compute $S$ given a
particular $P$, by search for a linking message block. This amounts
to a total running time of $2^{108}$. If partial knowledge of $P$
exists (as is the case when facing the challenge of predicting the
outcome of presidential elections when MD5 is
used~\cite{Stevens2008}), the attack can be much faster.
In order to exploit dedicated collision-search attacks on SHA-1, a collision search which is faster than about $2^{55.5}$ would be needed. Such a fast collision search would need to find a pair $(m,m^*)$ such that $h_c(cv_1,m)=h_c(cv_2,m^*)$ where the attacker has little control over the chaining variables $cv_1$ and $cv_2$. Such an algorithm is not known to date.
John Kelsey, Tadayoshi Kohno - Herding Hash Functions and the Nostradamus Attack
- EUROCRYPT 4004:183-200,2006
- http://dx.doi.org/10.1007/11761679_12
BibtexAuthor : John Kelsey, Tadayoshi Kohno
Title : Herding Hash Functions and the Nostradamus Attack
In : EUROCRYPT -
Address :
Date : 2006
