1 | initial version |

Let $q = p^n$ for some prime $p$ and some integer $n \ge 1$, and let $F_q$ be the field with $q$ elements, and $F_p$ the field with $p$ elements.

Recall that $F_q$ is a vector space over the prime field $F_p$, and that for any generator $z$ of $F_q$ as a field extension of $F_p$, the family $(1, z, ..., z^{n-1})$ is a basis of $F_q$ as a vector space over $F_p$.

One way to build a list of random nonzero elements in $F_q$ is to pick coefficients $a_0$, ..., $a_{n-1}$ at random, all between $0$ and $p-1$, but not all zero, and to take the element $\sum a_k z^k$ in $F_q$.

One way to pick such a collection of coefficients is to pick an integer at random between $1$ and $q-1$ and to let $a_k$ be its $k$-th digit in base $p$.

Choose $p$ and $n$, define $q = p^n$, and let $F$ be the finite field with $q$ elements.

```
sage: p = 5
sage: n = 3
sage: q = p^n
sage: F = GF(q)
```

Choose $m$, the number of random elements to pick.

```
sage: m = 10
```

Produce a list of length $m$ of random nonzero elements in $F$.

```
sage: L = [F(ZZ.random_element(1, q).digits(base=p)) for _ in range(m)]
```

`r = ZZ.random_element(1, q)`

picks a random integer between $1$ and $q - 1$`d = r.digits(base=p)`

gives the list of its digits in base $p$`u = F(d)`

turns this list into a field element, seen a polynomial in $z$ (where $z$ is the generator of $F_q$ as a field extension of $F_p$) with coefficients given by the list.

2 | No.2 Revision |

Let $q = p^n$ for some prime $p$ and some integer $n \ge 1$, and let $F_q$ be the field with $q$ elements, and $F_p$ the field with $p$ elements.

Recall that $F_q$ is a vector space over the prime field $F_p$, and that for any generator $z$ of $F_q$ as a field extension of $F_p$, the family $(1, z, ..., z^{n-1})$ is a basis of $F_q$ as a vector space over $F_p$.

One way to build a list of random nonzero elements in $F_q$ is to pick coefficients $a_0$, ..., $a_{n-1}$ at random, all between $0$ and $p-1$, but not all zero, and to take the element $\sum a_k z^k$ in $F_q$.

One way to pick such a collection of coefficients is to pick an integer at random between $1$ and $q-1$ and to let $a_k$ be its $k$-th digit in base $p$.

Choose $p$ and $n$, define $q = p^n$, and let $F$ be the finite field with $q$ elements.

```
sage: p = 5
sage: n = 3
sage: q = p^n
sage: F = GF(q)
```

Choose $m$, the number of random elements to pick.

```
sage: m = 10
```

Produce a list of length $m$ of random nonzero elements in $F$.

```
sage: L = [F(ZZ.random_element(1, q).digits(base=p)) for _ in range(m)]
```

`r = ZZ.random_element(1, q)`

picks a random integer between $1$ and $q - 1$`d = r.digits(base=p)`

gives the list of its digits in base $p$`u = F(d)`

turns this list into a field element, seen a polynomial in $z$ (where $z$ is the generator of $F_q$ as a field extension of $F_p$) with coefficients given by the list.

Defining the finite field takes on the order of 0.1 ms.

It is probably worth defining $F$ once and for all, and then picking random
elements in $F$, rather than calling `GF(25)`

inside the loop, which spends
time initializing the finite field at each iteration of the loop.

Of course if you're looping only ten times it doesn't matter much.

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