# 13.4 Birthday Attacks on Signatures

Alice is going to sign a document electronically by using one of the signature schemes to sign the hash of the document. Suppose the hash function produces an output of 50 bits. She is worried that Fred will try to trick her into signing an additional contract, perhaps for swamp land in Florida, but she feels safe because the chance of a fraudulent contract having the same hash as the correct document is 1 out of ${2}^{50}$, which is approximately 1 out of ${10}^{15}$. Fred can try several fraudulent contracts, but it is very unlikely that he can find one that has the right hash. Fred, however, has studied the birthday attack and does the following. He finds 30 places where he can make a slight change in the document: adding a space at the end of a line, changing a wording slightly, etc. At each place, he has two choices: Make the small change or leave the original. Therefore, he can produce ${2}^{30}$ documents that are essentially identical with the original. Surely, Alice will not object to any of these versions. Now, Fred computes the hash of each of the ${2}^{30}$ versions and stores them. Similarly, he makes ${2}^{30}$ versions of the fraudulent contract and stores their hashes. Consider the generalized birthday problem with $r={2}^{30}$ and $N={2}^{50}$. The probability of a match is approximately

Therefore, it is very likely that a version of the good document has the same hash as a version of the fraudulent contract. Fred finds the match and asks Alice to sign the good version. He plans to append her signature to the fraudulent contract, too. Since they have the same hash, the signature would be valid for the fraudulent one, so Fred could claim that Alice agreed to buy the swamp land.

But Alice is an English teacher and insists on removing a comma from one sentence. Then she signs the document, which has a completely different hash from the document Fred asked her to sign. Fred is foiled again. He now is faced with the prospect of trying to find a fraudulent contract that has the same hash as this new version of the good document. This is essentially impossible.

What Fred did is called the birthday attack. Its practical implication is that you should probably use a hash function with output twice as long as what you believe to be necessary, since the birthday attack effectively halves the number of bits. What Alice did is a recommended way to foil the birthday attack on signature schemes. Before signing an electronic document, make a slight change.