add primality figure to doc
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@ -4435,7 +4435,7 @@ reduction can be written as a single function with the Comba technique it is muc
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calling the half precision multipliers, addition and division by $\beta$ algorithms.
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For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly
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shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms
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shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DHREF} and ElGamal \cite{ELGAMALREF}. In these algorithms
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primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in
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modular exponentiation to greatly speed up the operation.
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@ -4725,7 +4725,7 @@ one of the algorithms presented in ~REDUCTION~.
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Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm
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will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
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value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm
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value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec:modinv}}). If no inverse exists the algorithm
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terminates with an error.
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\begin{figure}[!h]
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@ -6142,8 +6142,9 @@ of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate
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The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be
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discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by
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$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range
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$3 \le q \le 100$.
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$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}) demonstrates the probability of success for the range $3 \le q \le 100$.
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FIGU,primality,Probability of successful trial division to detect non-primes
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At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to
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be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate
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@ -6332,6 +6333,18 @@ JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.''
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\bibitem[19]{JAVA}
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The Sun Java Website, \url{http://java.sun.com/}
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\bibitem[20]{LeVeque}
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William LeVeque, \textit{Fundamentals of Number Theory}, Dover Publications, 2014
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\bibitem[21]{ELGAMALREF}
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T. Elgamal, \textit{A public key cryptosystem and a signature scheme based on discrete logarithms}, {IEEE} Transactions on Information Theory, 1985, pp. 469-472
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\bibitem[22]{TOOM}
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D. Knuth, \textit{The Art of Computer Programming; Volume 2. Third Edition}, Addison-Wesley, 1997, pg. 294
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\bibitem[23]{POSIX1}
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The Open Group, \url{http://www.opengroup.org/austin/papers/posix_faq.html}, 2017
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\end{thebibliography}
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\input{tommath.ind}
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