Add Discrete Fourier Transform example

[plafer]

Implements the Discrete Fourier Transform over the input list. Basically, this is the non-efficient version of the Number Theoretic Transform (NTT).

You can verify the correctness of the program by comparing with the Python implementation:

# Adapted from https://www.geeksforgeeks.org/python-number-theoretic-transformation/
from sympy import ntt 
  
seq = [15, 21, 13, 44] 
  
prime_no = 2**64 - 2**32 + 1
  
# ntt 
transform = ntt(seq, prime_no) 
print ("NTT : ", transform)

# NTT :  [93, 18440270144950239235, 18446744069414584284, 6473924464345090]

Conceptually though, this Miden implementation is closer to

from math import log2

PRIME = 2**64 - 2**32 + 1
TWO_ADIC_ROOT_OF_UNITY = 7277203076849721926

def get_root_of_unity(n):
    power = (1 << (32 - n)) % PRIME
    return pow(TWO_ADIC_ROOT_OF_UNITY, power, PRIME)

def f_k(seq, k, alpha):
    alpha_k = pow(alpha, k, PRIME)
    return sum([(seq[i] * pow(alpha_k, i, PRIME)) for i in range(len(seq))]) % PRIME

def ntt(seq, alpha):
    return [f_k(seq, k, alpha) for k in range(len(seq))]

seq = [15, 21, 13, 44]
alpha = get_root_of_unity(int(log2(len(seq))))

print(ntt(seq, alpha))

[Dominik1999]

This is amazing, thanks a lot.

I did a first round, but I need more time to wrap my head around it

[Dominik1999]

Super nice example. I left a few comments but overall, its really nice. And I learned some things about the Discrete Fourier Transform (DFT).

One general comment, we now try to follow a common commenting style in MASM, see https://github.com/0xPolygonMiden/miden-base/blob/main/miden-lib/asm/faucets/basic.masm#L39C1-L39C1 where we always

• explain a function or line of instructions above it, i.e. # get max supply of this faucet. We assume it is stored at pos 3 of slot 1
• indicate the new stack below it, i.e., # => [max_supply, amount, tag, RECIPIENT, ...]

[Dominik1999]

Do you need to store n again here? Looks like you did that already in procedure main?

[plafer]

This store_array procedure was written to be self-contained (modulo the fact that it uses the global INPUT_ARR_LOC instead of receiving that address on the stack, but I got lazy); i.e. it can be used to store array of any length. So you could do (pseudocode)

# Storing arrays of different lengths
exec.store_array([4,1,2,3,4])
exec.store_array([3,1,2,3])

Basically, an array is [length, ele_0, ..., ele_{length-1}], and this mem_store stores the length (which could be different from call to call).

[sonarcloud]

Kudos, SonarCloud Quality Gate passed!   

0 Bugs
0 Vulnerabilities
0 Security Hotspots
0 Code Smells

No Coverage information
No Duplication information

[plafer]

@Dominik1999 Changed the code to follow the comment convention

[Dominik1999]

Looks good to me, thanks!