Opened 11 years ago

Closed 8 years ago

# Finite Field elements in terms of powers of a generator

Reported by: Owned by: aly.deines cpernet minor sage-duplicate/invalid/wontfix finite rings GF, finite field Jeroen Demeyer N/A

For large values of q, a prime power, GF(q) has elements represented as polynomials over a generator.

```sage: F.<a> = GF(2^8)
sage: a^10
a^6 + a^5 + a^4 + a^2
```

If you further want to compute in a polynomial ring over F, then the polynomials aren't very pretty as they are polynomials with polynomial coefficients.

```sage: R.<x> = F[]
sage: a^10*x+1
(a^6 + a^5 + a^4 + a^2)*x + 1
```

It would be nice to be able to be able to print and work with the elements as powers of the generator.

### comment:1 Changed 11 years ago by lmartel

But I expect it to be slow... How would you compute this efficiently?

```sage: K.<a> = GF(17**17)
sage: a + 1
a^24137569    # this would be the result in your representation
```

### comment:2 Changed 9 years ago by jdemeyer

• Milestone changed from sage-5.11 to sage-5.12

### comment:3 Changed 9 years ago by vbraun_spam

• Milestone changed from sage-6.1 to sage-6.2

### comment:4 Changed 8 years ago by vbraun_spam

• Milestone changed from sage-6.2 to sage-6.3

### comment:5 Changed 8 years ago by vbraun_spam

• Milestone changed from sage-6.3 to sage-6.4

### comment:6 Changed 8 years ago by jdemeyer

• Description modified (diff)

### comment:7 Changed 8 years ago by jdemeyer

• Component changed from group theory to finite rings
• Owner changed from joyner to cpernet

### comment:8 Changed 8 years ago by jdemeyer

• Milestone changed from sage-6.4 to sage-duplicate/invalid/wontfix
• Reviewers set to Jeroen Demeyer
• Status changed from new to needs_review

This requires discrete log computations, which is too inefficient in general.

Close as "wontfix".

### comment:9 Changed 8 years ago by jdemeyer

• Status changed from needs_review to positive_review

### comment:10 Changed 8 years ago by vbraun

• Resolution set to wontfix
• Status changed from positive_review to closed
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