By Arndt J.

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If r is a generator, then every element in Fp different from 0 is equal to some power re (1 ≤ e < p) of r and its order is R/e. To test whether r is a primitive n-th root of unity in Fp one does not need to check rk = 1 for all k < n. It suffices to do the check for exponents k that are prime factors of n. This is because the order of any element divides the maximal order. 1 (Primitive root modulo p) Return a primitive root in Fp function primroot(p) { if p==2 then return 1 f[] := distinct_prime_factors(p-1) for r:=2 to p-1 { x := TRUE foreach q in f[] { if r**((p-1)/q)==1 then x:=FALSE } if x==TRUE then return r } error("no primitive root found") // p cannot be prime !

Cc] Transposing the array before the column pass in order to avoid the copying of the columns to extra scratch space will do good for the performance in most cases. The transposing back at the end of the routine can be avoided if a backtransform will follow9 , the backtransform must then be called with R and C swapped. The generalization to higher dimensions is straight forward. 10 The matrix Fourier algorithm (MFA) The matrix Fourier algorithm10 (MFA) works for (composite) data lengths n = R C.

A thorough discussion can be found in [35]. cc] 8 cf. 1 Definition of the HT The Hartley transform (HT) is defined like the Fourier transform with ‘cos + sin’ instead of ‘cos +i · sin’. e. symmetry is, like for the FT, conserved. 1 radix 2 FHT algorithms Decimation in time (DIT) FHT For a sequence a of length n let X 1/2 a denote the sequence with elements ax cos π x/n + ax sin π x/n (this is the ‘shift operator’ for the Hartley transform). 8) H [a] H [a]n = 49 CHAPTER 3. cc] The procedure hartley_shift replaces element ck of the input sequence c by ck cos(π k/n) + cn−k sin(π k/n).

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