1. Professor A G Constantinides 1 General Transforms Let be orthogonal, period N Define So that

2. Professor A G Constantinides 2 General Transforms Determine conditions to be satisfied by so that Let Then

3. Professor A G Constantinides 3 General Transforms Thus To support circular convolution 1) and real

4. Professor A G Constantinides 4 General Transforms 2) 3) Since fundamental period is N 4)

5. Professor A G Constantinides 5 Number Theoretic Transforms Thus in a complex field are the N roots of unity and In an integer field we can write and use Fermat's theorem whereis prime and is a primitive root Euler's totient function can be used to generalise as

6. Professor A G Constantinides 6 Number Theoretic Transforms Fermat's Theorem: Consider Reduce mod P to produce Since we have or and since there are no other unknown factors

7. Professor A G Constantinides 7 Number Theoretic Transforms Alternatively (perhaps simpler) For not multiples of P expanded in bionomial form produces multiples of P except for the terms Thus

8. Professor A G Constantinides 8 Number Theoretic Transforms Now, if the total number of bracketed terms is for this argument less than P say a, then for one has ie and

9. Professor A G Constantinides 9 Number Theoretic Transforms For example for P=7 the quantity a, known as the primitive root, will be one of the following {2,3,4,5,6} Thus for a=2 we have We note further that

10. Professor A G Constantinides 10 Number Theoretic Transforms Thus we have And hence Thus only real numbers are involved in the computation. Moreover, the kernel is a power of 2