General Transforms

1
General Transforms

• Let be orthogonal, period N
• Define
• So that

2
General Transforms

• Determine conditions to be satisfied by
• so that
• Let
• Then

3
General Transforms

• Thus
• To support circular convolution
• 1)
• and real

4
General Transforms

• 2)
• 3) Since fundamental period is N
• 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

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
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
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
Number Theoretic Transforms

• For example for P7 the quantity a, known as the
  primitive root, will be one of the following
  2,3,4,5,6
• Thus for a2 we have
• We note further that

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