From 2307cc9d4d34ab5b9f6de7ad7d7d52d13e731cf0 Mon Sep 17 00:00:00 2001 From: Daira Hopwood Date: Mon, 6 Feb 2017 01:22:20 +0000 Subject: [PATCH] Delete redundant "The notation ..." in Notation section. Signed-off-by: Daira Hopwood --- protocol/protocol.tex | 44 +++++++++++++++++++++---------------------- 1 file changed, 22 insertions(+), 22 deletions(-) diff --git a/protocol/protocol.tex b/protocol/protocol.tex index f06bc07..89933e1 100644 --- a/protocol/protocol.tex +++ b/protocol/protocol.tex @@ -900,12 +900,12 @@ one valid \nullifier, and so attempting to spend a \note twice would reveal the \nsection{Notation} -The notation $\bit$ means the type of bit values, i.e.\ $\setof{0, 1}$. +$\bit$ means the type of bit values, i.e.\ $\setof{0, 1}$. -The notation $\Nat$ means the type of nonnegative integers. $\PosInt$ +$\Nat$ means the type of nonnegative integers. $\PosInt$ means the type of positive integers. $\Rat$ means the type of rationals. -The notation $x \typecolon T$ is used to specify that $x$ has type $T$. +$x \typecolon T$ is used to specify that $x$ has type $T$. A cartesian product type is denoted by $S \times T$, and a function type by $S \rightarrow T$. An argument to a function can determine other argument or result types. @@ -921,25 +921,25 @@ written as subscripts, e.g.\ if $x \typecolon X$, $y \typecolon Y$, and $f \typecolon X \times Y \rightarrow Z$, then an invocation of $f(x, y)$ can also be written $f_x(y)$. -The notation $\typeexp{T}{\ell}$, where $T$ is a type and $\ell$ is an integer, +$\typeexp{T}{\ell}$, where $T$ is a type and $\ell$ is an integer, means the type of sequences of length $\ell$ with elements in $T$. For example, $\bitseq{\ell}$ means the set of sequences of $\ell$ bits. -The notation $\length(S)$ means the length of (number of elements in) $S$. +$\length(S)$ means the length of (number of elements in) $S$. -The notation $T \subseteq U$ indicates that $T$ is an inclusive subset or subtype of $U$. +$T \subseteq U$ indicates that $T$ is an inclusive subset or subtype of $U$. $\byteseqs$ means the set of bit sequences constrained to be of length a multiple of 8 bits. -The notation $\hexint{}$ followed by a string of \textbf{boldface} hexadecimal +$\hexint{}$ followed by a string of \textbf{boldface} hexadecimal digits means the corresponding integer converted from hexadecimal. -The notation $\ascii{...}$ means the given string represented as a +$\ascii{...}$ means the given string represented as a sequence of bytes in US-ASCII. For example, $\ascii{abc}$ represents the byte sequence $[\hexint{61}, \hexint{62}, \hexint{63}]$. -The notation $a..b$, used as a subscript, means the sequence of values +$a..b$, used as a subscript, means the sequence of values with indices $a$ through $b$ inclusive. For example, $\AuthPublicNew{\allNew}$ means the sequence $[\AuthPublicNew{\mathrm{1}}, \AuthPublicNew{\mathrm{2}}, ...\,\AuthPublicNew{\NNew}]$. @@ -948,55 +948,55 @@ this specification uses 1-based indexing and inclusive ranges, notwithstanding the compelling arguments to the contrary made in \cite{EWD-831}.) -The notation $\range{a}{b}$ means the set or type of integers from $a$ through +$\range{a}{b}$ means the set or type of integers from $a$ through $b$ inclusive. -The notation $\listcomp{f(x) \for x \from a \upto b}$ means the sequence +$\listcomp{f(x) \for x \from a \upto b}$ means the sequence formed by evaluating $f$ on each integer from $a$ to $b$ inclusive, in ascending order. Similarly, $\listcomp{f(x) \for x \from a \downto b}$ means the sequence formed by evaluating $f$ on each integer from $a$ to $b$ inclusive, in descending order. -The notation $a\,||\,b$ means the concatenation of sequences $a$ then $b$. +$a\,||\,b$ means the concatenation of sequences $a$ then $b$. -The notation $\concatbits(S)$ means the sequence of bits obtained by +$\concatbits(S)$ means the sequence of bits obtained by concatenating the elements of $S$ viewed as bit sequences. If the elements of $S$ are byte sequences, they are converted to bit sequences with the \emph{most significant} bit of each byte first. -The notation $\sorted(S)$ means the sequence formed by sorting the elements +$\sorted(S)$ means the sequence formed by sorting the elements of $S$. -The notation $\GF{n}$ means the finite field with $n$ elements, and +$\GF{n}$ means the finite field with $n$ elements, and $\GFstar{n}$ means its group under multiplication. $\GF{n}[z]$ means the ring of polynomials over $z$ with coefficients in $\GF{n}$. -The notation $a \mult b$ means the result of multiplying $a$ and $b$. +$a \mult b$ means the result of multiplying $a$ and $b$. This may refer to multiplication of integers, rationals, or finite field elements according to context. -The notation $a^b$, for $a$ an integer or finite field element and +$a^b$, for $a$ an integer or finite field element and $b$ an integer, means the result of raising $a$ to the exponent $b$. -The notation $a \bmod q$, for $a \typecolon \Nat$ and $q \typecolon \PosInt$, +$a \bmod q$, for $a \typecolon \Nat$ and $q \typecolon \PosInt$, means the remainder on dividing $a$ by $q$. -The notation $a \xor b$ means the bitwise-exclusive-or of $a$ and $b$, +$a \xor b$ means the bitwise-exclusive-or of $a$ and $b$, and $a \band b$ means the bitwise-and of $a$ and $b$. These are defined either on integers or bit sequences according to context. -The notation $\vsum{i=1}{\mathrm{N}} a_i$ means the sum of $a_{\allN{}}$.\; +$\vsum{i=1}{\mathrm{N}} a_i$ means the sum of $a_{\allN{}}$.\; $\vxor{i=1}{\mathrm{N}} a_i$ means the bitwise exclusive-or of $a_{\allN{}}$. The binary relations $<$, $\leq$, $=$, $\geq$, and $>$ have their conventional meanings on integers and rationals, and are defined lexicographically on sequences of integers. -The notation $\floor{x}$ means the largest integer $\leq x$. +$\floor{x}$ means the largest integer $\leq x$. $\ceiling{x}$ means the smallest integer $\geq x$. -The notation $\bitlength(x)$, for $x \typecolon \Nat$, means the smallest integer +$\bitlength(x)$, for $x \typecolon \Nat$, means the smallest integer $\ell$ such that $2^\ell > x$. The symbol $\bot$ is used to indicate unavailable information or a failed decryption.