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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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Source:  OpenStax, Automatic generation of prime length fft programs. OpenStax CNX. Sep 09, 2009 Download for free at http://cnx.org/content/col10596/1.4
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