# 0.12 Appendix: a matlab program for generating prime length fft  (Page 3/3)

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```function x = Kcrot(p,e,K,x) % Kronecker product of Cyclotomic Reduction Operations.% x = (G(p(1)^e(1)) kron ... kron G(p(K)^(K)))^t*x % (transpose)% p : p = [p(1),...,p(K)];% e : e = [e(1),...,e(K)];a = (p-1).*((p).^(e-1)); r = a; % r(i) = number of rows of G(i)c = 2*a-1; % c(i) = number of columns of G(i) m = 1;n = prod(r); for i = 1:Kn = n / r(i); x = IcrotI(p(i),e(i),m,n,x);m = m * c(i); end``` ```function y = IcrotI(p,e,m,n,x) % y = (eye(m) kron G(p^e)^t kron eye(n))*x% (transpose) a = (p-1)*(p^(e-1));c = a; r = 2*a-1;y = zeros(r*m*n,1); v = 0:n:(r-1)*n;u = 0:n:(c-1)*n; for i = 0:m-1for j = 0:n-1 y(v+i*r*n+j+1) = crot(p,e,x(u+i*c*n+j+1));end end``` ```function y = crot(p,e,x) % y = crot(p,x)% cyclotomic reduction matrix (transpose) % length(x) == 2*n-1% length(y) == n % where n = (p-1)*(p^(e-1))n = (p-1)*(p^(e-1)); y = zeros(2*n-1,1);if p == 2 n = p^(e-1);y(1:n) = x; y(n+1:2*n-1) = -x(1:n-1);else y(1:n) = x;L = p^(e-1); y(n+1:n+L) = -x(1:L);a = L; for k = 2:p-1y(n+1:n+L) = y(n+1:n+L) - x(a+1:a+L); a = a + L;end b = 2*n-1 - p*(p^(e-1));y(p*L+1:p*L+b) = x(1:b); end```

The following programs tell the programs for code generation relevant information about the bilinear forms for cyclotomic convolution.Specifically, they indicates the linear convolution out of which these cyclotomic convolution are composed, and thedimensions of the corresponding matrices. See the appendix Bilinear Forms for Linear Convolution .

```function [d,r,c,Q,Qt] = A_data(n)% A : A matrix in bilinear form for cyclotomic convolution % d : linear convolution modules used% r : rows % c : columns% Q : Q(i) = cost associated with D(d(i)) % Qt : Qt(i) = cost associated with D(d(i))'if n == 2, d = [1];elseif n == 4, d = [2];elseif n == 8, d = [2 2];elseif n == 16, d = [2 2 2];elseif n == 3, d = [2];elseif n == 9, d = [2 3];elseif n == 27, d = [2 3 3];elseif n == 5, d = [2 2];elseif n == 7, d = [2 3];end r = []; c = []; Q = []; Qt = [];for k = 1:length(d) [rk, ck, Qk, Qtk]= D_data(d(k)); r = [r rk]; c = [c ck]; Q = [Q Qk]; Qt = [Qt Qtk];end``` ```function [r,c,Q,Qt] = D_data(d);% D : D matrix in bilinear form for linear convolution % r : rows% c : columns % Q : cost associated with D(d)% Qt : cost associated with D(d)' if d == 1, r = 1; c = 1; Q = 0; Qt = 0;elseif d == 2, r = 3; c = 2; Q = 1; Qt = 2; elseif d == 3, r = 5; c = 3; Q = 7; Qt = 9;end``` ```function [f,r,c] = C_data(p,e)% f : length of linear convolution% r : rows % c : columnsf = prod((p-1).*(p.^(e-1))); % (Euler Totient Function)r = 2*f-1; c = F_data(f);``` ```function c = F_data(n) % c : columns of F matrixif n == 1, c = 1; elseif n == 2, c = 3;elseif n == 4, c = 9; elseif n == 8, c = 27;elseif n == 3, c = 5; elseif n == 6, c = 15;elseif n == 18, c = 75; end```

## Programs for inverse transpose reduction operations

```function x = itKRED(P,E,K,x) % x = itKRED(P,E,K,x);% (inverse transpose) % P : P = [P(1),...,P(K)]; % E : E = [E(K),...,E(K)]; for i = 1:Ka = prod(P(1:i-1).^E(1:i-1)); c = prod(P(i+1:K).^E(i+1:K));p = P(i); e = E(i);for j = e-1:-1:0 x(1:a*c*(p^(j+1))) = itRED(p,a,c*(p^j),x(1:a*c*(p^(j+1))));end end``` ```function y = itRED(p,a,c,x) % y = itRED(p,a,c,x);% (inverse transpose) y = zeros(a*c*p,1);for i = 0:c:(a-1)*c for j = 0:c-1A = x(i*p+j+1); for k = 0:c:c*(p-2)A = A + x(i*p+j+k+c+1); endy(i+j+1) = A; for k = 0:c:c*(p-2)y(i*(p-1)+j+k+a*c+1) = p*x(i*p+j+k+1) - A; endend endy = y/p;```

## Programs for permutations

The permutation of Equation 18 from Preliminaries is implemented by `pfp` . It calls the function `pfp2I` . The transpose is implemented by `pfpt` and it calls `pfpt2I` .

```function x = pfp(n,K,x) % x = P(n(1),...,n(K)) * x% n = [n(1),...,n(K)];% length(x) = prod(n(1),...,n(K)) a = prod(n);s = 1; for i = K:-1:2a = a / n(i); x = pfp2I(a,n(i),s,x);s = s * n(i); end``` ```function y = pfp2I(a,b,s,x) % y = kron(P(a,b),I(s)) * x;% length(x) = a*b*s n = a * b;y = zeros(n*s,1); k1 = 0;k2 = 0; for k = 0:n-1i1 = s * (k1 + b * k2); i2 = s * k;for i = 1:s y(i1 + i) = x(i2 + i);end k1 = k1 + 1;k2 = k2 + 1; if k1>= b k1 = k1 - b;end if k2>= a k2 = k2 - a;end end``` ```function x = pfpt(n,K,x) % x = P(n(1),...,n(K))' * x% (tanspose) % n = [n(1),...,n(K)]; % length(x) = prod(n(1),...,n(K))% a = prod(n); a = n(1);s = prod(n(2:K)); for i = 2:Ks = s / n(i); x = pfpt2I(a,n(i),s,x);a = a * n(i); end``` ```function y = pfpt2I(a,b,s,x) % y = P(a,b)' kron I(s) * x;% (transpose) % length(x) = a*b*sn = a * b; y = zeros(n*s,1);k1 = 0; k2 = 0;for k = 0:n-1 i1 = s * (k1 + b * k2);i2 = s * k; for i = 1:sy(i2 + i) = x(i1 + i); endk1 = k1 + 1; k2 = k2 + 1;if k1>= b k1 = k1 - b;end if k2>= a k2 = k2 - a;end end```

The following Matlab programs implement Rader's permutation and its transpose.They require the primitive root to be passed to them as an argument.

```function y = rp(p,r,x) % Rader's Permutation% p : prime % r : a primitive root of p% x : length(x) == p a = 1;y = zeros(p,1); y(1) = x(1);for k = 2:p y(k) = x(a+1);a = rem(a*r,p); end``` ```function y = rpt(p,r,x) % Rader's Permutation% (transpose) % p : prime% r : a primitive root of p % x : length(x) == pa = 1; y = zeros(p,1);y(1) = x(1); for k = 2:py(a+1) = x(k); a = rem(a*r,p);end``` ```function [R, R_inv] = primitive_root(N)% function [R, R_inv] = primitive_root(N)% Ivan Selesnick % N is assumed to be prime. This function returns R,% the smallest primitive root of N, and R_inv, the % inverse of R modulo N.R = 'Not Found'; m = 0:(N-2);for x = 1:(N-1) if ( 1:(N-1) == sort(rem2(x,m,N)) )R = x; breakend endR_inv = 'Not Found'; for x = 1:Nif rem(x*R,N) == 1 R_inv = x;break endend```

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