Rewritings due to feedback

thesis/sections/notation.tex

@@ -2,8 +2,7 @@
 
 \subsubsection{General Notation}
 
-% TODO: Notation mit residual ring und finite field abklären.
-For an integer n, $\field{n}$ is defined as the residual ring $\mathbb{Z}/n\mathbb{Z}$. $a \randomsample A$ denotes sampling the element $a$ from a non-empty set $A$ uniformly at random. $\assign$ denotes a deterministic assignment of a variable. $\{0,1\}^n$ is a bitstring of length n, while $\{0,1\}^*$ denotes a finite bitstring of arbitrary length. $(x,y)$ is a tuple of the two elements $x$ and $y$. $\{x,y\}$ is a set of the elements $x$ and $y$. At the beginning of a game a set is initialized to be the empty set $\{\}$. $\sum$ denotes a table and $\sum[x]$ denotes the value of the table at position $x$. Each position of the table is uninitialized at the beginning of the game. An uninitialized position in the table is denoted with the bottom symbol $\bot$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ is called negligible if there exists a $N \in \mathbb{N}$ so that for all polynomials $p$: $\forall n \geq N: f(n) < \frac{1}{p(n)}$ is true. All algorithms are probabilistic polynomial time (ppt) unless stated otherwise. $o \randomassign \adversary{A}(I)$ denotes running the algorithm $\adversary{A}$ with input $I$ with uniform random coins and $o$ describing its output. If $\adversary{A}$ has additionally access to an oracle $O$ this is denoted as $o \randomassign \adversary{A}^{O(\inp)}(I)$. A security game consists of a main procedure and optionally some oracle procedures. When the game is played, the main procedure is run and adversary $\adversary{A}$ is given some inputs and access to the oracle procedures. Based on the output of the adversary $\adversary{A}$ and its oracle calls, the main procedure outputs $1$ or $0$ depending on whether the adversary $\adversary{A}$ won the game.
+For an integer n, $\field{n}$ is defined as the residual ring $\mathbb{Z}/n\mathbb{Z}$. $a \randomsample A$ denotes sampling the element $a$ from a non-empty finite set $A$ uniformly at random. $\assign$ denotes a deterministic assignment of a variable. $\{0,1\}^n$ is the set of all bitstrings of length n, while $\{0,1\}^*$ denotes the set of finite bitstring of arbitrary length. $(x,y)$ is a tuple of the two elements $x$ and $y$. $\{x,y\}$ is a set of the elements $x$ and $y$. At the beginning of a game a set is initialized to be the empty set $\{\}$. $\sum$ denotes a table and $\sum[x]$ denotes the value of the table at position $x$. Each position of the table is uninitialized at the beginning of a game. An uninitialized position in the table is denoted with the bottom symbol $\bot$. A function $f: \mathbb{N} \rightarrow \mathbb{R}$ is called negligible if for all polynomials $p$ there exists a $N \in \mathbb{N}$ so that $\forall n \geq N: f(n) < \frac{1}{p(n)}$ is true. $\pset{S} \in f(x)$ denotes the set $\pset{S}$ of outputs of $f$ given $x$ as input.  All algorithms are probabilistic polynomial time (ppt) unless stated otherwise. $o \randomassign \adversary{A}(I)$ denotes running the algorithm $\adversary{A}$ with input $I$ with uniform random coins and $o$ describing its output. If $\adversary{A}$ has additionally access to an oracle $O$ this is denoted as $o \randomassign \adversary{A}^{O(\inp)}(I)$. A security game consists of a main procedure and optionally some oracle procedures. When a game is played, the main procedure is run and adversary $\adversary{A}$ is given some inputs and access to the oracle procedures. Based on the output of the adversary $\adversary{A}$ and its oracle calls, the main procedure outputs $1$ or $0$ depending on whether the adversary $\adversary{A}$ won the game. The message space of the signature scheme is defined as $\messagespace$.
 
 \subsubsection{Algebraic Notation}