# What are differences between $E(F_p)$ and $E(Z_p)$?

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 .

• Same thing, just difference in notation. – mikeazo Jun 6 '14 at 1:41
• 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 '14 at 2:07
• 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 '14 at 2:35

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.