A136415 Numbers n such that a type-3 Gaussian normal basis over GF(2^n) exists.

4, 6, 12, 14, 20, 22, 46, 52, 54, 60, 70, 76, 92, 94, 116, 124, 126, 140, 166, 174, 180, 182, 204, 206, 214, 220, 230, 236, 244, 252, 262, 276, 284, 286, 292, 294, 302, 332, 340, 350, 356, 364, 372, 374, 390, 404, 412, 430, 460, 484, 494, 510, 516, 526, 532

Offset: 1

Author

Joerg Arndt, Mar 31 2008

Keywords

• nonn

Comments

A type-t Gaussian normal basis exists for GF(2^n) if p=n*t+1 is prime and gcd(n,(p-1)/ord(2 mod p))==1.

Type-1 basis correspond to sequence A071642, type-2 basis to A054639.

Examples

			12 is in the list because 3*12+1=37 is prime and the index of 2 mod 37 (==36/ord(2 mod 37)==1, 2 is a generator mod 37) is coprime to 12.
		

Links

• Joerg Arndt, Mar 31 2008, Table of n, a(n) for n = 1..201
• Joerg Arndt, Matters Computational (The Fxtbook), section 42.9 "Gaussian normal bases", pp.914-920

Crossrefs

Cf. A071642, A054639.

Programs

PARI

gauss_test(n, t)=
{ /* test whether a type-t Gaussian normal basis exists for GF(2^n) */
  local( p, r2, g, d );
  p = t*n + 1;
  if ( !isprime(p), return( 0 ) );
  if ( p<=2, return( 0 ) );
  r2 = znorder( Mod(2, p) );
  d = (p-1)/r2;
  g = gcd(d, n);
  return ( if ( 1==g, 1, 0) );
}
/* generate this sequence: */
t=3;ct=1;for(n=1,10^7,if(gauss_test(n,t), print1(n,", ");ct+=1;if(ct>200,break())))