FFT
... Given N-bit integers (N ≥ 64) I and J, compute IJ. Assume: we can multiply words of O(log N) bits in constant time. Setup: Find a prime p=cn+1 that can be represented in one word, and set m=(log p)/3, so that we can view I and J as n-length vectors of m-bit words. Finding a primitive root of unity. ...
... Given N-bit integers (N ≥ 64) I and J, compute IJ. Assume: we can multiply words of O(log N) bits in constant time. Setup: Find a prime p=cn+1 that can be represented in one word, and set m=(log p)/3, so that we can view I and J as n-length vectors of m-bit words. Finding a primitive root of unity. ...
Lecture 1-3: Abstract algebra and Number theory
... (i) g(x) ≡ h(x) (mod f (x)) if and only if g(x) and h(x) leaves the same remainder when divided by f (x). (ii) g(x) ≡ g(x) (mod f (x)). (iii) If g(x) ≡ h(x) (mod f (x)) then h(x) ≡ g(x) (mod f (x)). iv) If g(x) ≡ h(x) (mod f (x)) and h(x) ≡ s(x) (mod f (x)), then g(x) ≡ s(x) (mod f (x)). v) If g(x) ...
... (i) g(x) ≡ h(x) (mod f (x)) if and only if g(x) and h(x) leaves the same remainder when divided by f (x). (ii) g(x) ≡ g(x) (mod f (x)). (iii) If g(x) ≡ h(x) (mod f (x)) then h(x) ≡ g(x) (mod f (x)). iv) If g(x) ≡ h(x) (mod f (x)) and h(x) ≡ s(x) (mod f (x)), then g(x) ≡ s(x) (mod f (x)). v) If g(x) ...
Finite Fields
... If φ is a ring homomorphism from a ring R onto a ring S then the factor ring R/kerφ and the ring S are isomorphic by the map r + kerφ 7→ φ(r). We can use mappings to transfer a structure from an algebraic system to a set without structure. Given a ring R, a set S and a bijective map φ : R → S, we ca ...
... If φ is a ring homomorphism from a ring R onto a ring S then the factor ring R/kerφ and the ring S are isomorphic by the map r + kerφ 7→ φ(r). We can use mappings to transfer a structure from an algebraic system to a set without structure. Given a ring R, a set S and a bijective map φ : R → S, we ca ...