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-57,8 +57,9 @@ systems. \section{Mathematical and Cryptographic Preliminaries} Throughout this paper, let $\mathbb{N}$ denote the non-negative integers including zero, i.e. $\mathbb{N} = \{0,1,2,\ldots\}$. +Further, let $\mathbb{Z}$ denote the integers $\mathbb{Z} = \{\ldots,-2,-1,0,1,2,\ldots\}$. One of the most fundamental notions in mathematics is without doubt -the term \emph{divisor}, we say for $a,c\in\mathbb{N}$, +the term \emph{divisor}, we say for $a,c\in\mathbb{Z}$, that \emph{$a$ divides $c$}, and write $a|c$, if there is a number $b\in\mathbb{Z}$ such that $c = a\cdot b$. Every number $n\in\mathbb{N}$ has at least two divisors, a \emph{prime number} $p\in\mathbb{N}$ is a number $p$ @@ -176,9 +177,22 @@ $$ \begin{itemize} \item $\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p$; \item $\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$; + \item $\left(\frac{a^2}{p}\right) = 1$; + \item If $a\equiv b\pmod p$ then $\left(\frac{a}{p}\right) = \left(\frac{b}{p}\right)$; \item $\sum_{a=0}^{p-1}\limits\left(\frac{a}{p}\right) = 0$; \item $\left(\frac{p}{q}\right) = (-1)^{(p-1)(q-1)/4} \left(\frac{q}{p}\right)$; - \item $\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$. + \item $\left(\frac{-1}{p}\right) = \begin{cases} + 1\, \qquad & p\equiv 1\pmod 4;\\ + -1\, \qquad & p\equiv 3\pmod 4; + \end{cases}$ + \item $\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$; + \end{itemize} + In particular, the last point specializes to + \begin{itemize} + \item $\left(\frac{2}{p}\right) = \begin{cases}1\,\qquad & \text{if $p\equiv 1$ or $p\equiv 7\pmod 8$};\\ + -1\,\qquad & \text{if $p\equiv 3$ or $p\equiv 5\pmod 8$}. + \end{cases}$ + \end{itemize} \end{lem} @@ -187,7 +201,8 @@ $$ \subsubsection*{The Discrete Logarithm problem} Consider an odd prime $p$. Let $b\in\mathbb{Z}_p$ and $0\leq x\leq p-1$. We consider the equality $a\equiv b^x\pmod p$, and call $x$ the \emph{discrete -logarithm} (or \emph{index}) of $a$ with respect to the basis $b$.\\ +logarithm} (or \emph{index}) of $a$ with respect to the basis $b$. It is indicated +by $x={\rm ind}_b(a)$.\\ The Diffie-Hellmann key exchange algorithm relies on the assumption, that given $a^x$ and $a^y$ (without knowing $x$ and $y$), the expression $a^{xy}$ can not be efficiently computed, @@ -195,7 +210,13 @@ i.e. the discrete logarithm problem is not effectively solvable. Effective solva context means that there is an algorithm which proceeds more effective than the brute-force method, which proceeds by computing $b^k\pmod p$ until $a\equiv b^x\pmod p$ is reached.\\ -TODO: Baby Step Giant Step Algorithm +We will now describe the Baby-Step/Giant-Step algorithm for computing the discrete logarithm. +Suppose we are given an odd prime $p$, an element $g\in\mathbb{Z}_p$ and $a = a_0\in\mathbb{Z}$. +The algorithm proceeds by first compiling a table with entries $(i,t_i)$ where $t_i = g^i \pmod p$. +The table consists of $m := \lceil \sqrt{p-1}\rceil$ entries from zero to $m-1$. +In the next step, one computes $g^{-m}\pmod p$. If $a$ is in the second row of the table, +on reads of the index (in the first row), otherwise it computes $a_{i+1} = a_i g^{-m}\pmod p$. +At some stage between $i = 0$ and $m-1$ we find the index $j$ and return as solution $x=i\cdot m + j$. \subsection{The RSA cryptosystem} @@ -250,7 +271,6 @@ $c^d \equiv c^{a_0}\cdot (c^{2})^{a_1} \cdot \ldots (c^{2^m})^{a_m}$. But some of the coefficients $a_k$ might be zero, and hence contribute 1 to the product. Hence, we are led to the following algorithm: \begin{algorithm}[H] - \begin{algorithmic}[1] \Require{$c\in\mathbb{Z}$, $d\in\mathbb{N}$} \Ensure{$c^d$} @@ -304,6 +324,58 @@ y \leq y' \text{ and } $$ The termination of the loop is a consequence of $\lfloor n/2^k\rfloor \rightarrow 0$ after finitely many steps, i.e. there exists $k\in\mathbb{N}$ such that $\lfloor n/2^k\rfloor = 0$. + +\subsection{Computing Integer Roots} +Given a number $x\in\mathbb{R}^+$, the \emph{square-root} $y := \sqrt{x}\in\mathbb{R}$ is implicit +defined by $y^2 = x$. To not leave the field of real numbers (and introducing complex numbers), we +restrict the domain to $\mathbb{R}^+$. However, in general, $x\in\mathbb{R}^+$ has two roots, +$y = \pm \sqrt{x}$.\\ + +Computing the \emph{integer-square-root} amounts to the following: Given a non-negative integer $a\in\mathbb{N}^+$, +usually its square root is either an integer or irrational. The integer-square-root is defined as the non-negative +integer $x\in\mathbb{N}^+$ that fulfills $x\leq \sqrt{a} a$ and we terminate the algorithm with +$x^2\leq a$ (as it holds in each step of the loop), and we conclude +$$ +y > a \Leftrightarrow (x^2+z) > a \Leftrightarrow x^2+2x+1 > a \Leftrightarrow (x+1)^2 > a, +$$ +hence we have shown $x^2\leq a < (x+1)^2$, and taking square roots shows the claim. \appendix \section{Listing of the developed Theory} {\scriptsize \verbatiminput{../numbertheory.txt}} -- 2.1.4