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Input: Any number $n \in \mathbb{Z}^+$ that can be represented in the form of $n = 2^a + b,\ |b|= c $. output: YES if $n$ is prime , else NO .

Now, length of binary input is $\log(a) + O(1)$ which otherwise would be order $a$. What would be the complexity of such problem? If we use AKS algorithm in straight forward way we would not get polynomial time algorithm, I think. Can we define new problems by tweaking the standard problems in P to make them harder in this way i.e. by restricting the input in certain way.

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The succinct version of problems are exponentially harder (e.g. Succinct-SAT is NExpTime-complete). Yours is a restricted case of succinct version of Prime. – Kaveh Dec 30 '12 at 6:14
Thanks, I was not aware that succinct version of problems is already well explored area. – DurgaDatta Dec 30 '12 at 6:26
You are welcome. ps: You may want to have a look at Russell Impagliazzo's answer to the padding question. – Kaveh Dec 30 '12 at 6:28
@Kaveh make this an answer ? – Suresh Venkat Jan 2 '13 at 16:50

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