Technion - Computer Science Department - Tehnical Report CS0578 ...
TECHNION - Israel Institute of Technology
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
Computer Science
Departme~t
GENERALIZATIONS OF THE NOkMAL BASIS THEOREM OF FINITE FIELDS
by
N.H. Bshouty and G. Seroussi Technical Renort #578 July 1989
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
GENERALIZAnONS OF THE NORMAL BASIS THEOREM OF FINITE FIELDS
Nader H. Bshoury •
and
Gadiel Seroussi ••
ABSTRACT We present a combinatorial characterization of sets of integers (rO,rl, ... ,TII-d, with
OSri~"_2,
"
such that a'O,({1, ... ,a'·-1 fonn a basis of GF(q") over GF(q) for some ae GF'(q"). We use this characterization to prove the following generalization of the nonnal basis theorem fpr finite -fields of characteristic two: Let Ao.Al; ... ,A.,.-1 be integers in the range >..0
~ (J denote the set of solutions
~ of (20) for a given
a, and let
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
7.
I.,emma 5: Con~der the transfonnation T:S" ~S" defined by (~T)(i)
If ~ e
~O'
for some a, then
Proof: Since
~e
c1>a,
~T ~
= ~(i+m) + m,
~e
S",
(21)
OSl Sn-l.
e c1>a. satisfies (20). Substituting i +m for i in the sum at the left hand side of
(20), and recalling'that indices are taken modulo n, we obtain 11-1
LA;+mqi+"'++(i+m) s;;" s(cr)
mod (q"-I).
(22)
i=iJ
Using the definition of T, and recalling that A;.- =A;, 0Si Sn -I, it follows from (22) that ,,-1
LA;qi+(.T)(i) ;..0
ED
s(cr)
mod (q"-1).
(23)
Hence, ~T e ~a. •
Lemma 6: For any ~e SII' let G.
= (~,~T,~T2, . .. ).
Then, IG.I
= i'
for some integer h,
OSh Sk.
Proof: >From the definition of T in (21), and from the fact that n=2" m, it follows that ~ T t for all ~ e S".
=~
Therefore, by a standard group-theoretic argument, the least positive integer g.
satisfying ~ T'. = ~ must divide 2". Hence, 1G.I = g. = 2" for some ~h Sk .•
Lemma 7: I c1> I is odd.
Proof: It follows from the result of Lemma 5 and from the arguments in the proof of Lemma 6 that
G.
the
transformation' T
= [~,~T,~T2, . .. ,~T'·-\
induces
a
partition
of c1>
into
disjoint
orbits
of the
form
If ~T~, then, by Lemma 6, g. is a nontrivial power of two, and G.
contributes an even number of pennutations to c1>. Hence, it remains to show that there is an odd number of pennutations ~ in c1> that satisfy ~ T==4l (i.e. g .=1). We claim that the set of all such ~ is precisely B". Since by Lemma 4(iii)' IB" I is odd, this would suffice to prove Lemma 7. To prove the
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
8.
claim, we first notice that if ~T=4l then ~ satisfies property (PI). Hence, «i)+i
= ~(i+m)+(i+m), and,
since A;=A;..... , OSiSn-l, we have II-I
",-I
LAiql+;(I)
= 21: LAiqi++(l),
(24)
1=0
i=O
and II~'
s(cr) = LA;q2a(i
mod",)
= 21:
1=0
"'~
LA; q 2o(1).
(25)
1=0
for all cre S",. Now, since ~e
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
Computer Science
Departme~t
GENERALIZATIONS OF THE NOkMAL BASIS THEOREM OF FINITE FIELDS
by
N.H. Bshouty and G. Seroussi Technical Renort #578 July 1989
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
GENERALIZAnONS OF THE NORMAL BASIS THEOREM OF FINITE FIELDS
Nader H. Bshoury •
and
Gadiel Seroussi ••
ABSTRACT We present a combinatorial characterization of sets of integers (rO,rl, ... ,TII-d, with
OSri~"_2,
"
such that a'O,({1, ... ,a'·-1 fonn a basis of GF(q") over GF(q) for some ae GF'(q"). We use this characterization to prove the following generalization of the nonnal basis theorem fpr finite -fields of characteristic two: Let Ao.Al; ... ,A.,.-1 be integers in the range >..0
~ (J denote the set of solutions
~ of (20) for a given
a, and let
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
7.
I.,emma 5: Con~der the transfonnation T:S" ~S" defined by (~T)(i)
If ~ e
~O'
for some a, then
Proof: Since
~e
c1>a,
~T ~
= ~(i+m) + m,
~e
S",
(21)
OSl Sn-l.
e c1>a. satisfies (20). Substituting i +m for i in the sum at the left hand side of
(20), and recalling'that indices are taken modulo n, we obtain 11-1
LA;+mqi+"'++(i+m) s;;" s(cr)
mod (q"-I).
(22)
i=iJ
Using the definition of T, and recalling that A;.- =A;, 0Si Sn -I, it follows from (22) that ,,-1
LA;qi+(.T)(i) ;..0
ED
s(cr)
mod (q"-1).
(23)
Hence, ~T e ~a. •
Lemma 6: For any ~e SII' let G.
= (~,~T,~T2, . .. ).
Then, IG.I
= i'
for some integer h,
OSh Sk.
Proof: >From the definition of T in (21), and from the fact that n=2" m, it follows that ~ T t for all ~ e S".
=~
Therefore, by a standard group-theoretic argument, the least positive integer g.
satisfying ~ T'. = ~ must divide 2". Hence, 1G.I = g. = 2" for some ~h Sk .•
Lemma 7: I c1> I is odd.
Proof: It follows from the result of Lemma 5 and from the arguments in the proof of Lemma 6 that
G.
the
transformation' T
= [~,~T,~T2, . .. ,~T'·-\
induces
a
partition
of c1>
into
disjoint
orbits
of the
form
If ~T~, then, by Lemma 6, g. is a nontrivial power of two, and G.
contributes an even number of pennutations to c1>. Hence, it remains to show that there is an odd number of pennutations ~ in c1> that satisfy ~ T==4l (i.e. g .=1). We claim that the set of all such ~ is precisely B". Since by Lemma 4(iii)' IB" I is odd, this would suffice to prove Lemma 7. To prove the
Technion - Computer Science Department - Tehnical Report CS0578 - 1989
8.
claim, we first notice that if ~T=4l then ~ satisfies property (PI). Hence, «i)+i
= ~(i+m)+(i+m), and,
since A;=A;..... , OSiSn-l, we have II-I
",-I
LAiql+;(I)
= 21: LAiqi++(l),
(24)
1=0
i=O
and II~'
s(cr) = LA;q2a(i
mod",)
= 21:
1=0
"'~
LA; q 2o(1).
(25)
1=0
for all cre S",. Now, since ~e
