# Collision resistant functions - definition

Let $$f$$ be a collision resistant function i.e. it is computationally impossible to find $$x_0, x_1$$ such that $$f(x_0) = f(x_1)$$. If a computationally bounded adversary demonstrates that he knows some $$x_0$$ and $$f(x_0)$$, in what sense is he unable to determine $$x_1$$?

My instinct is that he can do no better than choosing a random guess from the domain of $$f$$ but is this how collision resistance is defined? Moreover, if $$f$$ has a trapdoor, is the definition still the same i.e. collision resistance implies that either the adversary knows the trapdoor or he can do no better than pick a random element from the domain of $$f$$?

I apologize that this is a bit of a yes/no question although if the answer is no, then there is more to say!

EDIT: Alternatively, what is the mathematical statement that captures the idea that there is no efficient algorithm to find a collision?

Let the range $$Y$$ of the function $$f$$ have size $$N$$ A collision resistant function cannot be better than an ideal function which maps uniformly into $$Y$$ since collision probability is minimized by the uniform distribution. Thus, given $$(x_0,f(x_0))$$ the number $$k$$ of random points $$x_i\neq x_0$$ that must be tried before the probability that $$f(x_i)=f(x_0)$$ is significant, say at least $$1/2$$ is $$k\approx \sqrt{\ln 2 N}$$ by the birthday paradox.

If necessary see @fgrieu's detailed post on the mathematics of the birthday paradox here.

• Thank you. Could you also comment on the following: When one says there is no efficient algorithm to find a collision, what is the mathematical definition this corresponds to? – user1936752 Oct 11 '19 at 10:06
• Let the output space have bitlength $b,$ hence $b=\log_2 N.$ The algorithm in the answer needs approximately $\sqrt{N}=2^{b/2}$ queries before finding a collision with probability $1/2.$ This quantity is exponential in the input size (bitlength) $b,$ hence inefficient. If it could be found with $b^c$ trials, for constant $c$, it would have polynomial complexity in the bitlength, hence efficient. – kodlu Oct 11 '19 at 12:18
• Ah I see - so no matter what algorithmthe attacker uses, the complexity is the same as a random guess attack? Thank you! – user1936752 Oct 11 '19 at 12:29
• Please keep in mind that if the adversary has full control about the definition of f and the choose of $x_0$ and $x_i$ this requirement is very hard to fulfill since the adversary can define any two $x_0$ and $x_1$ such that $h(x_0) = h(x_1)$. There has to be a mechanism which makes it hard for the adversary to define the functions f like that. – Martin Kromm Oct 11 '19 at 13:48
• @MartinKromm, I can choose $f$ instead of the adversary and then this is fine, no? The point is then that $x_1$ is indistinguishable from a random string (in the domain of $f$) for the adversary. – user1936752 Oct 11 '19 at 14:43

Defining a hash function formally is not an easy task. There are even entire papers covering that formalization, especially if there is a requirement that the function should be unkeyed (see for the keyed variant below).

The most popular variant to formalize hash functions are the use of choosing a hash function chosen from a family of hash functions HashFamily(KeyGen, Hash):

• Given security parameter $$1^n$$, algorithm KeyGen outputs a key k.
• Given key k, a family of hashes $$H := \{h_0, ..., h_{2^n - 1}\}$$ and a message m, algorithm Hash outputs $$h_k(m)$$ with $$h_k \in H$$.

Now consider the following game Col:

• The adversary A is given H and a random key k.
• The adversary A outputs $$m_1$$ and $$m_2$$.
• The adversary A wins if $$h_k(m_1) = h_k(m_2)$$ and $$m_1 \neq m_2$$.

Collision resistant means that $$\forall PPT A: Pr[Col(A) = 1] \leq negl(n)$$.

The above game implies that the adversary A can compute $$Hash(k, H, \cdot)$$ by himself. However, since the adversary is computationally bounded, he can only query a limited number of messages (if the adversary is unbounded, then the adversary always wins the game unless $$h_\cdot$$ is injective of course).

PPT A means probabilistic polynomial time algorithm. That means that given some randomness r, algorithm A computes x using any strategy in $$O(n^z)$$ (with z being constant and independent of the input size).

I would also recommend you reading Introduction to Modern Cryptography by Katz & Lindell. There is the topic explained in more detail.