code-examples.tex

\section{Simple Coding Examples}

\outlinecomment{Get cocori to contribute here.}

\subsection{Entropy}

From the fantastic website
RosettaCode.org~\footnote{\url{http://rosettacode.org/wiki/Entropy}},
we have a simple example in Python of calculating the entropy in bits
per byte:

%\begin{widetext}
%\begin{verbatim}
\begin{lstlisting}[language=Python]
import math
from collections import Counter

def entropy(s):
    p, lns = Counter(s), float(len(s))
    return -sum( count/lns * math.log(count/lns, 2) for count in p.values())

entropy("1223334444")
\end{lstlisting}
%\end{verbatim}
%\end{widetext}

(This function can actually calculate entropy of strings of other
types besides characters, as well.)

\subsection{Birthday Paradox/Collision Probability}

Some ad hoc Mathematica code for dinking around with some of the
numbers:

% \begin{verbatim}
\begin{lstlisting}[language=Mathematica]
NoCollisionProbability[n_, D_] := N[Factorial[D]/Factorial[D - n]/D^n]

NoCollisionProbability[23, 365]
0.492703

NoCollisionProbability[2^10, 2^20]
0.606728

Stirling[n_] := Sqrt[2*\[Pi]*n]*(n/E)^n

NoCollisionProbabilityApprox[n_, D_] := 
 N[Stirling[D]/Stirling[D - n]/D^n]

NoCollisionProbabilityApprox[2^10, 2^20]
0.6067282267199

NoCollisionProbabilityApprox[2^12, 2^24]
0.606580027339
\end{lstlisting}
% \end{verbatim}
(That last example took several minutes on my laptop.  Be very careful
running the above code with larger numbers, it's very easy to start a
computation that won't complete in your lifetime, and Mathematica
isn't as friendly about stopping computations in a reasonable state as
one might like.)

A good analysis of the birthday bound appears in Sec. 2.2 of the
SWEET32 paper by Bhargavan and Leurent.

The square root limit as $n$ and $d$ grow large, in Mathematica:

% \begin{verbatim}
\begin{lstlisting}[language=Mathematica]
N[Sqrt[E]]
1.64827

N[1 - 1/Sqrt[E]]
0.393469
\end{lstlisting}
% \end{verbatim}

\subsection{QKD}

BB84 in Qiskit:
\url{https://github.com/qiskit-community/may4_challenge_exercises/blob/master/ex03/Challenge3_BB84_Solutions.ipynb}