Probability of getting a collision using chosen plaintext attacks

hasin

11/21/23, 5:19 PM

For university I am doing a piece of coursework right now. This question is focusing on CPA and collisions using CPA.

Question:

I have attempted to answer part 3, but am not very confident in the answer. If anyone could please take the time to look at the question and look at my answer to let me know if I am going in the right sort of direction that would be greatly appreciated.

My answer:

From the description of the encryption scheme in this question I have come to the conclusion that if $2^u$ queries are needed to have a 50% chance of predicting whether b is 0 or 1, this is because after $2^u$ queries the random nonce will have repeated itself at least once. However, due to the fact that the oracle has a 50% chance of returning the encryption of P0 or P1 $2^u$ queries only gives you a 50% chance of guessing the value of b.

As part 3 is only asking for the number of queries needed to correctly guess b with a probability greater than a half that means that $q > 2^u$. Then using the birthday paradox $q > \sqrt{2^u}$

Once again, thank you to anyone that takes the time out to respond to this.

0 + 1

birthday-attack

randomness

chosen-plaintext-attack

collision-resistance

nonce

fgrieu

11/21/23, 7:04 PM

The part of the would-be answer telling the random nonce must have repeated itself after $2^u$ queries is correct. Hint: what should be these $2^u$ queries and the handling of their results to make good use of that fact, and find $b$ with certainty, assuming $v\ge u$? Using the same strategy only with less queries, when does it succeed with certainty ? Note: if one reads Q3 literally, the correct answer is AT LEAST TWO, and the note in parenthesis is pointless; I guess Q3 really wants to ask how big $q$ should be to have nearly 50% (or more than 39%) chance to guess $b$ _with certainty_.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Probability of getting a collision using chosen plaintext attacks

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