Python | Number Theoretic Transformation

Number Theoretic Transform is a Fast Fourier transform theorem generalization. It is obtained by the replacement of e^(-2piik/N) with an nth primitive unity root. So this means, instead of the complex numbers C, use transform over the quotient ring Z/pZ. The theory is based on and uses the concepts of finite fields and number theory.

Number Theoretic Transform modulus need to be prime necessarily. But if it is prime, it makes things simpler. One can perform the NTT with a composite modulus. For modulus :

• n^th root of unity existence
• multiplicative inverse of n

A number-theoretic transform is basically a Fourier transform. Also, suppose that a Normal Discrete Fourier Transform is given and it can be done in matrix form by multiplying the data with a Fourier Matrix. Let us suppose N = 4. Then, the matrix can be –

[ 1   1    1    1  ]
[ 1   w   w^2  w^3 ]
[ 1  w^2  w^4  w^6 ]
[ 1  w^3  w^6  w^9 ]

sympy.discrete.transforms.ntt( ) :

It can Number Theoretic Transform (NTT) of the sequence.
It specializes over the Discrete Fourier Transform (DFT) quotient ring with Z/pZ for the prime na complexstead of acomplex numbers.

Automatically the sequence is padded with zero to the right because the radix-2 FFT requires the sample point number as a power of 2.

Parameters : 
-> seq       : [iterable] sequence on which DFT is to be applied.
-> prime no. : [Integer] prime modulus for NTT to perform on.

Returns : 
Number Theoretic Transform

Example 1 :

# import sympy
from sympy import ntt

# sequence
seq = [15, 21, 13, 44]

prime_no = 3 * 2**8 + 1

# ntt
transform = ntt(seq, prime_no)
print ("NTT : ", transform)

Output :

NTT :  [93, 114, 732, 659]

Example 2 :

# import sympy
from sympy import ntt

# sequence
seq = [153, 321, 133, 44, ]

# Prime modulus of the form (m * 2**k + 1)
prime_no = 3 * 2**8 + 1

# ntt
transform = ntt(seq, prime_no)
print ("NTT : ", transform)

Output :

NTT :  [651, 276, 690, 533]