The Sum Can Be Weaker Than Each Part

In this paper we study the security of summing the outputs of two

independent

hash functions, in an effort to increase the security of the resulting design, or to hedge against the failure of one of the hash functions. The exclusive-or (XOR) combiner

$$H_1(M) \oplus H_2(M)$$

is one of the two most classical combiners, together with the concatenation combiner

$$H_1(M) \Vert H_2(M)$$

. While the security of the concatenation of two hash functions is well understood since Joux’s seminal work on multicollisions, the security of the sum of two hash functions has been much less studied. The XOR combiner is well known as a good PRF and MAC combiner, and is used in practice in TLS versions 1.0 and 1.1. In a hash function setting, Hoch and Shamir have shown that if the compression functions are modeled as random oracles, or even

weak

random oracles (

i.e.

they can easily be inverted – in particular

$$H_1$$

and

$$H_2$$

offer no security),

$$H_1 \oplus H_2$$

is indifferentiable from a random oracle up to the birthday bound.

In this work, we focus on the preimage resistance of the sum of two narrow-pipe

$$n$$

-bit hash functions, following the Merkle-Damgård or HAIFA structure (the internal state size and the output size are both

$$n$$

bits).We show a rather surprising result: the sum of two such hash functions, e.g. SHA-512

$$\oplus $$

Whirlpool, can never provide

$$n$$

-bit security for preimage resistance. More precisely, we present a generic preimage attack with a complexity of

$$\tilde{O}(2^{5n/6})$$

. While it is already known that the XOR combiner is not preserving for preimage resistance (

i.e.

there might be

some

instantiations where the hash functions are secure but the sum is not), our result is much stronger: for

any

narrow-pipe functions, the sum is not preimage resistant.

Besides, we also provide concrete preimage attacks on the XOR combiner (and the concatenation combiner) when one or both of the compression functions are weak; this complements Hoch and Shamir’s proof by showing its tightness for preimage resistance.

Of independent interests, one of our main technical contributions is a novel structure to control

simultaneously

the behavior of independent hash computations which share the same input message. We hope that breaking the pairwise relationship between their internal states will have applications in related settings.