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I recently encountered a breakthrough in FHE crypto, which claims to have a literally quasi-linear time FHE without any "lambda" factor in the keys and no noise in the cipher-text.

This fully homomorphic encryption works like for addition like $E(m_1)·E(m_2) = E(m_1+m_2)$ and for multiplication like $E(m_1)^{m_2} = E(m_1·m_2)$ ; they say they achieved it using p-adic exponential.

My Question is: What is exactly p-adic exponential?

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    What you describe is not fully homomorphic. Notice that $m_2$ in the second equation is not encrypted.2012-12-02
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    Hasn't he defined it in P1.3? Were you looking for more than that?2012-12-04
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    A warning about the paper: it doesn't sound like the author knows much about crypto. There's the misrepresentation of FHE that Mike pointed out; in addition, he claims his system is Informationally Secure (which is impossible for any public key system); he claims that a message cannot be uniquely decrypted; if this were true, this would imply that someone with the private key cannot decrypt it. His claim of $O(n \log n)$ time is also bogus. I would approach this system only with extreme caution.2012-12-04

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