Cryptography Stack Exchange is a question and answer site for software developers, mathematicians and others interested in cryptography. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

While reading some papers, I found that sometimes $Z_p$ was used but sometimes $F_p$ was used ($p$ is prime). Usually, the author will only use their subgroups like this: choose a big prime factor of $p-1$ , denote $q$, and set $G=\langle g \rangle$.

My questions are :

  • Why do we only use the subgroup while $Z_p$ OR $F_p$? Or maybe we can choose a $p$ such that $p-1$ is prime and now if $Z_p^*$ or $F_p^*$ can be directly used?
  • What's the difference of using $Z_p$ and $F_p$?
share|improve this question

migrated from Dec 1 '13 at 19:41

This question came from our site for theoretical computer scientists and researchers in related fields.

When $p$ is prime, $\mathbb{F}^\times_p \approx \mathbb{Z}_{p-1}$. – Yuval Filmus Oct 7 '13 at 5:30
I do not think this question fits here, it's just basic terminology. And also it's hard to answer as it is because it lacks context and the notation $Z_p$ is ambiguous. In the biggest generality, $\mathbf{Z}/n\mathbf{Z}$ is a ring. Authors sometimes use $\mathbf{Z}_n$ to denote the addition group of that ring. When $n$ is a prime $p$, then $\mathbf{Z}/p\mathbf{Z}$ is isomorphic to the finite field of size $p$, $\mathbf{F}_p$. And to make it more confusing, sometimes authors use $\mathbf{Z}_p$ to denote the multiplication group of $\mathbf{F}_p$. – Sasho Nikolov Oct 7 '13 at 16:18
yes i want to migrate it to Cryptography beta but it still needs 9 flags remaining – T.B Oct 7 '13 at 23:59
If $p$ is prime (and not 2 or 3), then $p-1$ is even and not equals to 2 and therefore cannot be prime. – Maeher Dec 2 '13 at 8:11

You should read the Wikipedia article on finite fields. For each prime $p$ and for each $n >0$, there is a unique field of order $p^n$ (up to isomorphism). This field is usually denoted $F_{p^n}$. Now, for $n = 1$, the field $F_p$ can be identified with the set $\mathbf{Z}/p\mathbf{Z}$ of integers modulo $p$, which is also sometimes denoted $\mathbf{Z}_p$. However, the notation $\mathbf{Z}_p$ is also used in number theory to denote the ring of $p$-adic integers and this may create a potential confusion.

The multiplicative group of every finite field is cyclic, that is, is generated by a single element. This multiplicative group is often used in cryptography.

share|improve this answer
by your means , there is no difference between $Z_p$ and $F_p$ ? – T.B Oct 7 '13 at 13:23
another question : why is the subgroup of $Z_p$ or $F_p$ used, for example in the cryptography protocols while itself ? – T.B Oct 7 '13 at 13:36
If (and only if) $p$ is prime, then $\mathbb{Z}_p$ is isomorphic to $\mathbb{F}_p$. Most of the time $\mathbb{Z}_n$ is used if calculations are just done in $\mathbb{Z}$ modulo some integer $n$. Finite fields however, also exits with index $p^n$, which are not "just integers". Usually they are constructed as integer polynomials modulo a irreducible polynomial of degree $n$ (s.t. all elements of the field have a lower degree). The different notations only emphasize the underlying structure (and which attributes are important). – tylo Dec 2 '13 at 13:13

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.