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When I read some books about elliptic curve cryptography noticed that. sometimes symbolized elliptic curve over $F_p$ is $E(F_p)$ and sometimes symbolized elliptic curve over $Z_p$ is $E(Z_p)$.

I know p is prime number In both cases.

I wonder. Is there a difference between that symbols or they have same meaning .

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3  
Same thing, just difference in notation. –  mikeazo Jun 6 at 1:41
6  
The difference comes if you do ECC in an extension field; where the field size is $p^k$ for $k>1$. In that case, $Z_{p^k}$ would imply that you're doing your math modulo $p^k$, which would be incorrect (that's not a field). Instead, in that case $E(F_{p^k})$ would be the correct notation. Of course, nowadays we usually stick to prime fields; in that case both notations are equivalent. –  poncho Jun 6 at 2:07
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As a number theorist, I would say that $\mathbb{Z}_p$ is working over the $p$-adic integers, which shouldn't really happen in cryptography, so you may ignore my comment. :) –  BlackAdder Jun 6 at 2:35

1 Answer 1

Just adding the comments of poncho and blackadder here which I think should have just been posted as an answer (answer is community wiki in order not to cowardly get credit for this).

The difference comes if you do ECC in an extension field; where the field size is $p^k$ for k>1. In that case, $Z_{p^k}$ would imply that you're doing your math modulo $p^k$, which would be incorrect (that's not a field). Instead, in that case $E(F_{p^k})$ would be the correct notation. Of course, nowadays we usually stick to prime fields; in that case both notations are equivalent.

Note that as BlackAdder mentioned the correct answer might depend on the author of the paper, if the paper was written by a number theorist then there is a chance that $Z_p$ means the $p$-adic integers, and arithmetic in $Z$ modulo $p^k$ (i.e. the $Z_{p^k}$ in poncho his comment) would be denoted by $Z/{p^k}Z$ in such papers.

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