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What is a preimage and how do you find a preimage of a hash?

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In both mathematics and cryptography, given a function $H$ from set $A$ to set $B$, and an element $b$ in $B$, a preimage of $b$ by $H$ is any $a$ in $A$ such that $H(a)=b$.

In cryptography, a public function $H$ from set $A$ to finite set $B$ is:

  • First-preimage-resistant when for a given random $b$ in $B$, it is hard to exhibit a preimage of $b$, that is, $a$ in $A$ with $H(a)=b$.
  • Second-preimage-resistant when for a given random $a_0$ in $A$, it is hard to exhibit another preimage of $b=H(a_0)$, that is, $a$ in $A$ with $a\ne a_0$ and $H(a)=H(a_0)$.

A preimage can in principle be found by trying various values of $a$ in $A$ (other that $a_0$ for second-preimage), and computing $H(a)$ until it matches $b$ (the given $b$ for first-preimage, or $b=H(a_0)$ computed from the given $a_0$ for second-preimage). Depending on the definition of $H$, there can be better methods.

A common design goal of practical cryptographic hash functions is that the expected effort to find a preimage (of either kind) is not much less than $|B|/2$ times the effort for computing $H(a)$ once, where the notation $|B|$ designates the number of elements in the set $B$. When $B$ is the set of exactly $n$-bit bitstrings $\{0,1\}^n$ (as is common for cryptographic hashes) the quantity $|B|/2$ becomes $2^{n-1}$.

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Since $f$ (hash function) is hard to invert, all you can do is try random inputs until you succeed.

If the hash function output is $n$ bits long, and the hash is strong (well approximated by a random function) your probability of success is only about $$\frac{q}{2^n}$$ after $q$ trials given output $y_0$ and checking whether $f(X_t)=y_0$ for random inputs $$X_1,\ldots, X_t,\ldots,X_q.$$

In the worst case you may need $q>2^n$ to succeed with probability one assuming $y_0$ is an actual output of the hash function since other outputs along the way will typically collide. Most likely $q\approx n 2^n$ will be needed which is the expected cover time for a balls and bins process (coupon collector).

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  • $\begingroup$ There are many conversion questions between primate problems but I did not find this one, interestingly. $\endgroup$ – kodlu Sep 22 '17 at 3:42

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