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?
To answer the question (and ignoring the fluff), the $p$-adic exponential is a partial function on the $p$-adic numbers defined by the power series
$$ \exp(x) = \sum_{i=0}^{\infty} \frac{x^n}{n!} $$
This power series converges if and only if $x$ is a $p$-adic integer divisible by $p$. It satisfies many of the properties you'd expect. For example, $\exp(x+y) = \exp(x) \exp(y)$, and it has an inverse, the $p$-adic logarithm.
If you're not familiar with the $p$-adic numbers, we can approximate them with modular arithmetic. If $x$ is an integer such that $x \equiv 0 \pmod p$, then the above power series also makes sense in the arithmetic of integers modulo $p^n$. For an example with the $3$-adic logarithm:
$$\begin{align} \exp(3) &\equiv 1 + 3 + \frac{9}{2} + \frac{27}{6} + \cdots& \pmod{27} \\ &\equiv 1 + 3 + \frac{9}{2} + \frac{9}{2} + 0& \pmod{27} \\ &\equiv 1 + 3 + 9 \cdot 14 + 9 \cdot 14 & \pmod{27} \\ &\equiv 13& \pmod{27} \end{align}$$
Note the remaining terms (indicated by $\cdots$) are all divisible by $27$, which is why I replaced withm with $0$ in the above calculation. And that $2$ and $14$ are inverses modulo $27$: i.e. $2 \cdot 14 \equiv 1 \pmod{27}$.