The birthday attack explained

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The birthday attack is the cryptographic attack type that cracks the algorithms of mathematics by finding matches in the hash function. The method relies upon the birthday paradox through which the chance of sharing one birthday by two people is quite higher than it appears. In the same way, the chance of collision detection is also higher within the target hash function, thereby enables the attacker to find similar fragments by the use of a few iterations. The attack is used mostly for abusing communication among the two parties. The nature of the attack is dependent upon the collisions found among random attacks as well as the degree of the permutation.

What is the Birthday Paradox Problem?

The probability theory describes the birthday paradox problem as if you carry the ‘n’ number of people in the room, there is the probability that some of them might have their birthdays on a similar day. But, one important point to consider is that we do not focus on the matching birth dates but rather two people sharing one birthday is the major point.
So, the birthday attacks utilize the approach of probability to minimize the difficulty in the matched collision and to get the approximate hash collision risk in a certain number. It also shows that discovering a particular hash collision is the most problematic thing instead of finding the matched hash collisions with similar values.

Calculating birthday paradox

Most people assume 183 as it is actually half of every possible birthday and it seems intuitive as well. However, it is not the game of intuition so, for the calculation, there are few assumptions to be made.

First of all, we will not consider the leap year as it will simplify the math and doesn’t change results. We will also consider that every birthday has an equal occurrence chance.
So, start with one person and then keep on adding people one at a time to show how this calculation is working. For the calculations, this is easy to calculate the probability that no other person will share the birthday. This probability will be subtracted from the other one derived that includes at least two people will share one birthday.

Chance of one match at least = 1 – probability of no match

For first person, there is 365/365 chance that there is no other shared birthday.

Now, add the second person. The first person has all chances of one birthday but the second person will have only 364/365 chance of not sharing similar day birthday. This goes:

Similarly, for the third person, the probability will be 363/365 of not sharing similar day birth day.

So, the pattern for the entire population will be:

 

When the graph is plotted in excel for the particular values, it shows birthday paradox problem answer.

When the probabilities are known, the answer to the birthday problem becomes 50.7% chance of people sharing people in total of 23 people group.

If the group size is increased, the probability will be reduced. Varying group size will have different effect on the probability.

 

How to prevent the Birthday attack

To prevent the birthday attack, there is the possibility that the length output for the hash function of the signature scheme can be selected large enough such that the chance of birthday attack computationally becomes impossible. 

Along with using the extended bit length, the signer can also prevent the attack if make some inoffensive but random changes to the document before it is signed and keep the contract copy under possession. Such that he can demonstrate within the court that the signature matches with the contract.

Final Thoughts

The birthday attack cracks the mathematics algorithm by its matching in the hash function. The birthday attack is best calculated with the probability theory. However, the attack can be prevented by increasing the bit length and if the signer makes some random changes within the document.

 

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