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