<Text-field style="Heading 1" layout="Heading 1">FFT mod p Background and code.</Text-field>

This section reviews how to compute N-th roots of unity mod p, the computation of the DFT mod p, and provides code to compute the FFT of a vector of elements in Z_p.

with(LinearAlgebra);

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with(numtheory);

Warning, the protected name order has been redefined and unprotected

First review how to access subvectors of a vector. This is needed to get the even and odd elements of a vector in the FFT.

a := Vector(8,(i)->i-1);

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

b := a[[seq(2*i+1,i=0..3)]];

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

c := a[[seq(2*i+2,i=0..3)]];

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

# Compute DFT using the Fast Fourier Transform # Inputs: N - an integer such that N = 2^k # a - a vector of size N with elements in Z_p # w - a primitive N-th root of unity in Z_p # p - an odd prime such that N|(p-1) # Output: A - a vector of size N with elements in Z_p such that A = DFT_N*a FFTmodp := proc(N,a,w,p)

local n,b,c,A,B,C,k,i;

if N = 1 then

return a;

else

n := N/2;

b := a[[seq(2*i+1,i=0..n-1)]];

c := a[[seq(2*i+2,i=0..n-1)]];

B := FFTmodp(n,b,w^2 mod p,p);

C := FFTmodp(n,c,w^2 mod p,p); A := Vector(N);

for k from 0 to n-1 do

A[k+1] := (B[k+1] + w^k*C[k+1]) mod p;

A[k+n+1] := (B[k+1] - w^k*C[k+1]) mod p;

end do; end if;

return A;

end;

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

# Construct the DFT matrix DFT_N is the N X N matrix whose (i,j) element = w^((i-1)*(j-1)), # where w is a primitive Nth root of unity in Z_p, p a prime with N|(p-1) DFT := proc(N,w,p)

Matrix(N,(i,j)->w^((i-1)*(j-1)) mod p);

end;

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

Test FFTmodp using p = 17 and N = 8. We will test the FFT by applying it to a vector with all 1's. As shown in class, the DFT of such a vector is a vector whose first element is N and all other elements are equal to 0. This is shown by direct computation of the DFT and then is verified with the FFT code.

We know by the primitive element theorem that there exists a primitive (p-1)st root of unity in Z_p. If \316\261 is a primitive element in Z_p then all of the non-zero elements of Z_p can be expressed as powers of \316\261. Alternatively the numbers 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 are distinct. The Maple command primroot in the numtheory package computes a primitive element in Z_p. Given a primitive element \316\261 in Z_p and a divisor d of p-1, then 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 is a primitive d-th root of unity.

p := 17;

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

N := 8;

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USJORigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjsvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkcvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUkjbW9HRiU2M1EjOj1GKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYoLyUncnNwYWNlR0ZbcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctSSNtbkdGJTY5USI4RihGMEYzRjYvRjpGOEY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ24vRmpuUSdub3JtYWxGKEZcbzcjLV9GKUksbXByaW50c2xhc2hHRig2JDcjPkkiTkdGKCIiKTcjRmVy

alpha := primroot(p);

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

Verify that \316\261 is a primitive element for Z_p.

seq(alpha^i mod p,i=0..p-1);

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

Construct a primitive N-th root of unity as a power of \316\261 and verify that it is a primitive N-th root of unity.

w := Power(alpha,(p-1)/N) mod p;

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

seq(w^i mod p,i=0..N);

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

Test FFT code.

a := Vector(N,1);

F8 := DFT(8,w,p);

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

A := MatrixVectorMultiply(F8,a) mod p;

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

A := FFTmodp(N,a,w,p);

Compute inverse DFT and inverse FFT.

wp := 1/w mod p;

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

w*wp mod p;

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

F8inv := 1/8 * DFT(8,wp,p) mod p;

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

MatrixMatrixMultiply(F8,F8inv) mod p;

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

When multipling A = F8*a by F8inv, we should get a back. I.E. F8inv * A = F8inv * F8 * a = a. Test this directly and then by using the FFT.

MatrixVectorMultiply(F8inv,A) mod p;

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

(1/N)*FFTmodp(N,A,wp,p) mod p;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2M1EiW0YnLyUlZm9ybUdRJ3ByZWZpeEYnLyUmZmVuY2VHUSV0cnVlRicvJSpzZXBhcmF0b3JHUSZmYWxzZUYnLyUnbHNwYWNlR1EudGhpbm1hdGhzcGFjZUYnLyUncnNwYWNlR0Y6LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjcvJShtYXhzaXplR1EpaW5maW5pdHlGJy8lKG1pbnNpemVHUSIxRicvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnLyUlc2l6ZUdRIzEyRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLUYjNiMtSSdtdGFibGVHRiQ2Ki1JJG10ckdGJDYjLUkkbXRkR0YkNiMtSSNtbkdGJDY5RkYvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHRlIvJSVib2xkR0Y3LyUnaXRhbGljR0Y3LyUqdW5kZXJsaW5lR0Y3LyUqc3Vic2NyaXB0R0Y3LyUsc3VwZXJzY3JpcHRHRjcvJStmb3JlZ3JvdW5kR0ZVLyUrYmFja2dyb3VuZEdGWC8lJ29wYXF1ZUdGNy8lK2V4ZWN1dGFibGVHRjcvJSlyZWFkb25seUdGNC8lKWNvbXBvc2VkR0Y3LyUqY29udmVydGVkR0Y3LyUraW1zZWxlY3RlZEdGNy8lLHBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR0ZPLyUqbWF0aGNvbG9yR0ZVLyUvbWF0aGJhY2tncm91bmRHRlgvJStmb250ZmFtaWx5R0Zjby8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSltYXRoc2l6ZUdGUkZobkZobkZobkZobkZobkZobkZobi1GLDYzUSJdRicvRjBRKHBvc3RmaXhGJ0YyRjVGOC9GPFEydmVyeXRoaW5tYXRoc3BhY2VGJ0Y9Rj9GQUZERkdGSUZLRk1GUEZTRlY=