General Transforms

1. General Transforms • Let be orthogonal, period N • Define • So that 1

2. General Transforms • Determine conditions to be satisfied by so that • Let • Then 2

3. General Transforms • Thus • To support circular convolution • 1) and real 3

4. General Transforms • 2) • 3) Since fundamental period is N • 4) 4

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 • where is prime and is a primitive root • Euler's totient function can be used to generalise as 5

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 6

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 7

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 8

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 9

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 10