%%% Compute the complement to one of the hybrid distribution cumulative %%%
%%% function defined in BLONDEAU-GERARD FSE 2011 i.e. P[C_k > tau]      %%%

%%% Parameters:
%%%    tau:  is the relative threshold 0 < tau < 1
%%%    q = p   if considering the  wrong  key candidate
%%%        p_* if considering the correct key candidate 
%%%    Ns:  number of samples ($N |\Delta_0|/2$)

function res=hybridcdf_comp(tau,q,Ns)

%%% this coefficient determines the interval where Poisson approximation is used
coeff = 3.0;

if ( tau < q - coeff*sqrt(q/Ns) )
  tau = (tau*Ns+0.5)/Ns;
  res = 1-exp(-Ns*Kullback_Bernoulli(tau,q)) * (q*sqrt(1-tau)/((q-tau)*sqrt(2*pi*tau*Ns)) + 1/sqrt(8*pi*tau*Ns));
else
  if ( tau > q + coeff*sqrt(q/Ns) )      
    tau = (tau*Ns+0.5)/Ns;
    res = exp(-Ns*Kullback_Bernoulli(tau,q)) * ((1-q)*sqrt(tau)/((tau-q)*sqrt(2*pi*(1-tau)*Ns)) + 1/sqrt(8*pi*tau*Ns));
  else
    res = 1-poisscdf(floor(tau*Ns)+1,q*Ns);
  end

end