Optimal normal bases

✍ Scribed by Shuhong Gao; Hendrik W. Lenstra

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 364 KB
• Volume: 2
• Category: Article
• ISSN: 0925-1022
• DOI: 10.1007/bf00125200

✦ Synopsis

Let K C L be a finite Galois extension of fields, of degree n. Let G be the Galois group, and let (act)o~G be a normal basis for L over K. An argument due to Mullin, Onyszchuk, Vanstone and W'flson (Discrete Appl. Math. 22 (1988/89), 149-161) shows that the matrix that describes the map x ~ otx on this basis has at least 2n -1 nonzero entries. If it contains exactly 2n -1 nonzero entries, then the normal basis is said to be optimal. In the present paper we determine all optimal normal bases. In the case that K is finite our result confirms a conjecture that was made by Mullin et al. on the basis of a computer search.