Miscellaneous stuff - libzahl - big integer library
(HTM) git clone git://git.suckless.org/libzahl
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---
(DIR) commit 16a001c8fe4e5ca99d5aafdd8ed02a35f09b6caa
(DIR) parent c98c0af42f56d0859d6f3f5e3743945906c681c8
(HTM) Author: Mattias Andrée <maandree@kth.se>
Date: Fri, 13 May 2016 04:38:09 +0200
Miscellaneous stuff
Signed-off-by: Mattias Andrée <maandree@kth.se>
Diffstat:
M Makefile | 6 +++++-
M TODO | 4 ++++
M doc/arithmetic.tex | 5 +++--
A doc/bit-operations.tex | 56 +++++++++++++++++++++++++++++++
M doc/libzahl.tex | 4 ++++
A doc/not-implemented.tex | 554 +++++++++++++++++++++++++++++++
A doc/number-theory.tex | 35 +++++++++++++++++++++++++++++++
A doc/random-numbers.tex | 28 ++++++++++++++++++++++++++++
M src/zmodmul.c | 1 +
M src/zmodpow.c | 2 ++
M src/zmodpowu.c | 5 +++--
M src/zpow.c | 2 ++
M src/zpowu.c | 5 +++--
13 files changed, 700 insertions(+), 7 deletions(-)
---
(DIR) diff --git a/Makefile b/Makefile
@@ -78,7 +78,11 @@ TEXSRC =\
doc/libzahls-design.tex\
doc/get-started.tex\
doc/miscellaneous.tex\
- doc/arithmetic.tex
+ doc/arithmetic.tex\
+ doc/bit-operations.tex\
+ doc/number-theory.tex\
+ doc/random-numbers.tex\
+ doc/not-implemented.tex
HDR_PUBLIC = zahl.h $(HDR_SEMIPUBLIC)
HDR = $(HDR_PUBLIC) $(HDR_PRIVATE)
(DIR) diff --git a/TODO b/TODO
@@ -4,6 +4,10 @@ It uses optimised division algorithm that requires that d|n.
Add zsets_radix
Add zstr_radix
+Can zmodpowu and zmodpow be improved using some other algorithm?
+Is it worth implementing precomputed optimal
+ addition-chain exponentiation in zpowu?
+
Test big endian
Test always having .used > 0 for zero
Test negative/non-negative instead of sign
(DIR) diff --git a/doc/arithmetic.tex b/doc/arithmetic.tex
@@ -186,8 +186,9 @@ can be expressed as a simple formula
\vspace{-1em}
\[ \hspace*{-0.4cm}
- a^b = \prod_{i = 0}^{\lceil \log_2 b \rceil}
- \left ( a^{2^i} \right )^{\left \lfloor {\displaystyle{b \over 2^i}} \hspace*{-1ex} \mod 2 \right \rfloor}
+ a^b =
+ \prod_{k \in \textbf{Z}_{+} ~:~ \left \lfloor {b \over 2^k} \hspace*{-1ex} \mod 2 \right \rfloor = 1}
+ a^{2^k}
\]
\noindent
(DIR) diff --git a/doc/bit-operations.tex b/doc/bit-operations.tex
@@ -0,0 +1,56 @@
+\chapter{Bit operations}
+\label{chap:Bit operations}
+
+TODO
+
+\vspace{1cm}
+\minitoc
+
+
+\newpage
+\section{Boundary}
+\label{sec:Boundary}
+
+TODO % zbits zlsb
+
+
+\newpage
+\section{Logic}
+\label{sec:Logic}
+
+TODO % zand zor zxor znot
+
+
+\newpage
+\section{Shift}
+\label{sec:Shift}
+
+TODO % zlsh zrsh
+
+
+\newpage
+\section{Truncation}
+\label{sec:Truncation}
+
+TODO % ztrunc
+
+
+\newpage
+\section{Split}
+\label{sec:Split}
+
+TODO % zsplit
+
+
+\newpage
+\section{Bit manipulation}
+\label{sec:Bit manipulation}
+
+TODO % zbset
+
+
+\newpage
+\section{Bit test}
+\label{sec:Bit test}
+
+TODO % zbtest
(DIR) diff --git a/doc/libzahl.tex b/doc/libzahl.tex
@@ -82,6 +82,10 @@ all copies or substantial portions of the Document.
\input doc/get-started.tex
\input doc/miscellaneous.tex
\input doc/arithmetic.tex
+\input doc/bit-operations.tex
+\input doc/number-theory.tex
+\input doc/random-numbers.tex
+\input doc/not-implemented.tex
\appendix
(DIR) diff --git a/doc/not-implemented.tex b/doc/not-implemented.tex
@@ -0,0 +1,554 @@
+\chapter{Not implemented}
+\label{chap:Not implemented}
+
+In this chapter we maintain a list of
+features we have choosen not to implement,
+but would fit into libzahl had we not have
+our priorities straight. Functions listed
+herein will only be implemented if there
+is shown that it would be overwhelmingly
+advantageous.
+
+\vspace{1cm}
+\minitoc
+
+
+\newpage
+\section{Extended greatest common divisor}
+\label{sec:Extended greatest common divisor}
+
+\begin{alltt}
+void
+extgcd(z_t bézout_coeff_1, z_t bézout_coeff_2, z_t gcd
+ z_t quotient_1, z_t quotient_2, z_t a, z_t b)
+\{
+#define old_r gcd
+#define old_s bézout_coeff_1
+#define old_t bézout_coeff_2
+#define s quotient_2
+#define t quotient_1
+ z_t r, q, qs, qt;
+ int odd = 0;
+ zinit(r), zinit(q), zinit(qs), zinit(qt);
+ zset(r, b), zset(old_r, a);
+ zseti(s, 0), zseti(old_s, 1);
+ zseti(t, 1), zseti(old_t, 0);
+ while (!zzero(r)) \{
+ odd ^= 1;
+ zdivmod(q, old_r, old_r, r), zswap(old_r, r);
+ zmul(qs, q, s), zsub(old_s, old_s, qs);
+ zmul(qt, q, t), zsub(old_t, old_t, qt);
+ zswap(old_s, s), zswap(old_t, t);
+ \}
+ odd ? abs(s, s) : abs(t, t);
+ zfree(r), zfree(q), zfree(qs), zfree(qt);
+\}
+\end{alltt}
+
+
+\newpage
+\section{Least common multiple}
+\label{sec:Least common multiple}
+
+\( \displaystyle{
+ \mbox{lcm}(a, b) = {\lvert a \cdot b \rvert \over \mbox{gcd}(a, b)}
+}\)
+
+
+\newpage
+\section{Modular multiplicative inverse}
+\label{sec:Modular multiplicative inverse}
+
+\begin{alltt}
+int
+modinv(z_t inv, z_t a, z_t m)
+\{
+ z_t x, _1, _2, _3, gcd, mabs, apos;
+ int invertible, aneg = zsignum(a) < 0;
+ zinit(x), zinit(_1), zinit(_2), zinit(_3), zinit(gcd);
+ *mabs = *m;
+ zabs(mabs, mabs);
+ if (aneg) \{
+ zinit(apos);
+ zset(apos, a);
+ if (zcmpmag(apos, mabs))
+ zmod(apos, apos, mabs);
+ zadd(apos, mabs, apos);
+ \}
+ extgcd(inv, _1, _2, _3, gcd, apos, mabs);
+ if ((invertible = !zcmpi(gcd, 1))) \{
+ if (zsignum(inv) < 0)
+ (zsignum(m) < 0 ? zsub : zadd)(x, x, m);
+ zswap(x, inv);
+ \}
+ if (aneg)
+ zfree(apos);
+ zfree(x), zfree(_1), zfree(_2), zfree(_3), zfree(gcd);
+ return invertible;
+\}
+\end{alltt}
+
+
+\newpage
+\section{Random prime number generation}
+\label{sec:Random prime number generation}
+
+TODO
+
+
+\newpage
+\section{Symbols}
+\label{sec:Symbols}
+
+\subsection{Legendre symbol}
+\label{sec:Legendre symbol}
+
+TODO
+
+
+\subsection{Jacobi symbol}
+\label{sec:Jacobi symbol}
+
+TODO
+
+
+\subsection{Kronecker symbol}
+\label{sec:Kronecker symbol}
+
+TODO
+
+
+\subsection{Power residue symbol}
+\label{sec:Power residue symbol}
+
+TODO
+
+
+\subsection{Pochhammer \emph{k}-symbol}
+\label{sec:Pochhammer k-symbol}
+
+\( \displaystyle{
+ (x)_{n,k} = \prod_{i = 1}^n (x + (i - 1)k)
+}\)
+
+
+\newpage
+\section{Logarithm}
+\label{sec:Logarithm}
+
+TODO
+
+
+\newpage
+\section{Roots}
+\label{sec:Roots}
+
+TODO
+
+
+\newpage
+\section{Modular roots}
+\label{sec:Modular roots}
+
+TODO % Square: Cipolla's algorithm, Pocklington's algorithm, Tonelli–Shanks algorithm
+
+
+\newpage
+\section{Combinatorial}
+\label{sec:Combinatorial}
+
+\subsection{Factorial}
+\label{sec:Factorial}
+
+\( \displaystyle{
+ n! = \left \lbrace \begin{array}{ll}
+ \displaystyle{\prod_{i = 0}^n i} & \textrm{if}~ n \ge 0 \\
+ \textrm{undefined} & \textrm{otherwise}
+ \end{array} \right .
+}\)
+\vspace{1em}
+
+This can be implemented much more efficently
+than using the naïve method, and is a very
+important function for many combinatorial
+applications, therefore it may be implemented
+in the future if the demand is high enough.
+
+
+\subsection{Subfactorial}
+\label{sec:Subfactorial}
+
+\( \displaystyle{
+ !n = \left \lbrace \begin{array}{ll}
+ n(!(n - 1)) + (-1)^n & \textrm{if}~ n > 0 \\
+ 1 & \textrm{if}~ n = 0 \\
+ \textrm{undefined} & \textrm{otherwise}
+ \end{array} \right . =
+ n! \sum_{i = 0}^n {(-1)^i \over i!}
+}\)
+
+
+\subsection{Alternating factorial}
+\label{sec:Alternating factorial}
+
+\( \displaystyle{
+ \mbox{af}(n) = \sum_{i = 1}^n {(-1)^{n - i} i!}
+}\)
+
+
+\subsection{Multifactorial}
+\label{sec:Multifactorial}
+
+\( \displaystyle{
+ n!^{(k)} = \left \lbrace \begin{array}{ll}
+ 1 & \textrm{if}~ n = 0 \\
+ n & \textrm{if}~ 0 < n \le k \\
+ n((n - k)!^{(k)}) & \textrm{if}~ n > k \\
+ \textrm{undefined} & \textrm{otherwise}
+ \end{array} \right .
+}\)
+
+
+\subsection{Quadruple factorial}
+\label{sec:Quadruple factorial}
+
+\( \displaystyle{
+ (4n - 2)!^{(4)}
+}\)
+
+
+\subsection{Superfactorial}
+\label{sec:Superfactorial}
+
+\( \displaystyle{
+ \mbox{sf}(n) = \prod_{k = 1}^n k^{1 + n - k}
+}\), undefined for $n < 0$.
+
+
+\subsection{Hyperfactorial}
+\label{sec:Hyperfactorial}
+
+\( \displaystyle{
+ H(n) = \prod_{k = 1}^n k^k
+}\), undefined for $n < 0$.
+
+
+\subsection{Raising factorial}
+\label{sec:Raising factorial}
+
+\( \displaystyle{
+ x^{(n)} = {(x + n - 1)! \over (x - 1)!}
+}\), undefined for $n < 0$.
+
+
+\subsection{Failing factorial}
+\label{sec:Failing factorial}
+
+\( \displaystyle{
+ (x)_n = {x! \over (x - n)!}
+}\), undefined for $n < 0$.
+
+
+\subsection{Primorial}
+\label{sec:Primorial}
+
+\( \displaystyle{
+ n\# = \prod_{\lbrace i \in \textbf{P} ~:~ i \le n \rbrace} i
+}\)
+\vspace{1em}
+
+\noindent
+\( \displaystyle{
+ p_n\# = \prod_{i \in \textbf{P}_{\pi(n)}} i
+}\)
+
+
+\subsection{Gamma function}
+\label{sec:Gamma function}
+
+$\Gamma(n) = (n - 1)!$, undefined for $n \le 0$.
+
+
+\subsection{K-function}
+\label{sec:K-function}
+
+\( \displaystyle{
+ K(n) = \left \lbrace \begin{array}{ll}
+ \displaystyle{\prod_{i = 1}^{n - 1} i^i} & \textrm{if}~ n \ge 0 \\
+ 1 & \textrm{if}~ n = -1 \\
+ 0 & \textrm{otherwise (result is truncated)}
+ \end{array} \right .
+}\)
+
+
+\subsection{Binomial coefficient}
+\label{sec:Binomial coefficient}
+
+\( \displaystyle{
+ {n \choose k} = {n! \over k!(n - k)!}
+ = {1 \over (n - k)!} \prod_{i = k + 1}^n i
+ = {1 \over k!} \prod_{i = n - k + 1}^n i
+}\)
+
+
+\subsection{Catalan number}
+\label{sec:Catalan number}
+
+\( \displaystyle{
+ C_n = \left . {2n \choose n} \middle / (n + 1) \right .
+}\)
+
+
+\subsection{Fuss–Catalan number}
+\label{sec:Fuss-Catalan number} % not en dash
+
+\( \displaystyle{
+ A_m(p, r) = {r \over mp + r} {mp + r \choose m}
+}\)
+
+
+\newpage
+\section{Fibonacci numbers}
+\label{sec:Fibonacci numbers}
+
+Fibonacci numbers can be computed efficiently
+using the following algorithm:
+
+\begin{alltt}
+ static void
+ fib_ll(z_t f, z_t g, z_t n)
+ \{
+ z_t a, k;
+ int odd;
+ if (zcmpi(n, 1) <= 1) \{
+ zseti(f, !zzero(n));
+ zseti(f, zzero(n));
+ return;
+ \}
+ zinit(a), zinit(k);
+ zrsh(k, n, 1);
+ if (zodd(n)) \{
+ odd = zodd(k);
+ fib_ll(a, g, k);
+ zadd(f, a, a);
+ zadd(k, f, g);
+ zsub(f, f, g);
+ zmul(f, f, k);
+ zseti(k, odd ? -2 : +2);
+ zadd(f, f, k);
+ zadd(g, g, g);
+ zadd(g, g, a);
+ zmul(g, g, a);
+ \} else \{
+ fib_ll(g, a, k);
+ zadd(f, a, a);
+ zadd(f, f, g);
+ zmul(f, f, g);
+ zsqr(a, a);
+ zsqr(g, g);
+ zadd(g, a);
+ \}
+ zfree(k), zfree(a);
+ \}
+
+ void
+ fib(z_t f, z_t n)
+ \{
+ z_t tmp, k;
+ zinit(tmp), zinit(k);
+ zset(k, n);
+ fib_ll(f, tmp, k);
+ zfree(k), zfree(tmp);
+ \}
+\end{alltt}
+
+\noindent
+This algorithm is based on the rules
+
+\vspace{1em}
+\( \displaystyle{
+ F_{2k + 1} = 4F_k^2 - F_{k - 1}^2 + 2(-1)^k = (2F_k + F_{k-1})(2F_k - F_{k-1}) + 2(-1)^k
+}\)
+\vspace{1em}
+
+\( \displaystyle{
+ F_{2k} = F_k \cdot (F_k + 2F_{k - 1})
+}\)
+\vspace{1em}
+
+\( \displaystyle{
+ F_{2k - 1} = F_k^2 + F_{k - 1}^2
+}\)
+\vspace{1em}
+
+\noindent
+Each call to {\tt fib\_ll} returns $F_n$ and $F_{n - 1}$
+for any input $n$. $F_{k}$ is only correctly returned
+for $k \ge 0$. $F_n$ and $F_{n - 1}$ is used for
+calculating $F_{2n}$ or $F_{2n + 1}$. The algorithm
+can be speed up with a larger lookup table than one
+covering just the base cases. Alternatively, a naïve
+calculation could be used for sufficiently small input.
+
+
+\newpage
+\section{Lucas numbers}
+\label{sec:Lucas numbers}
+
+Lucas numbers can be calculated by utilising
+{\tt fib\_ll} from \secref{sec:Fibonacci numbers}:
+
+\begin{alltt}
+ void
+ lucas(z_t l, z_t n)
+ \{
+ z_t k;
+ int odd;
+ if (zcmp(n, 1) <= 0) \{
+ zset(l, 1 + zzero(n));
+ return;
+ \}
+ zinit(k);
+ zrsh(k, n, 1);
+ if (zeven(n)) \{
+ lucas(l, k);
+ zsqr(l, l);
+ zseti(k, zodd(k) ? +2 : -2);
+ zadd(l, k);
+ \} else \{
+ odd = zodd(k);
+ fib_ll(l, k, k);
+ zadd(l, l, l);
+ zadd(l, l, k);
+ zmul(l, l, k);
+ zseti(k, 5);
+ zmul(l, l, k);
+ zseti(k, odd ? +4 : -4);
+ zadd(l, l, k);
+ \}
+ zfree(k);
+ \}
+\end{alltt}
+
+\noindent
+This algorithm is based on the rules
+
+\vspace{1em}
+\( \displaystyle{
+ L_{2k} = L_k^2 - 2(-1)^k
+}\)
+\vspace{1ex}
+
+\( \displaystyle{
+ L_{2k + 1} = 5F_{k - 1} \cdot (2F_k + F_{k - 1}) - 4(-1)^k
+}\)
+\vspace{1em}
+
+\noindent
+Alternatively, the function can be implemented
+trivially using the rule
+
+\vspace{1em}
+\( \displaystyle{
+ L_k = F_k + 2F_{k - 1}
+}\)
+
+
+\newpage
+\section{Bit operation}
+\label{sec:Bit operation unimplemented}
+
+\subsection{Bit scanning}
+\label{sec:Bit scanning}
+
+Scanning for the next set or unset bit can be
+trivially implemented using {\tt zbtest}. A
+more efficient, although not optimally efficient,
+implementation would be
+
+\begin{alltt}
+ size_t
+ bscan(z_t a, size_t whence, int direction, int value)
+ \{
+ size_t ret;
+ z_t t;
+ zinit(t);
+ value ? zset(t, a) : znot(t, a);
+ ret = direction < 0
+ ? (ztrunc(t, t, whence + 1), zbits(t) - 1)
+ : (zrsh(t, t, whence), zlsb(t) + whence);
+ zfree(t);
+ return ret;
+ \}
+\end{alltt}
+
+
+\subsection{Population count}
+\label{sec:Population count}
+
+The following function can be used to compute
+the population count, the number of set bits,
+in an integer, counting the sign bit:
+
+\begin{alltt}
+ size_t
+ popcount(z_t a)
+ \{
+ size_t i, ret = zsignum(a) < 0;
+ for (i = 0; i < a->used; i++) \{
+ ret += __builtin_popcountll(a->chars[i]);
+ \}
+ return ret;
+ \}
+\end{alltt}
+
+\noindent
+It requires a compiler extension, if missing,
+there are other ways to computer the population
+count for a word: manually bit-by-bit, or with
+a fully unrolled
+
+\begin{alltt}
+ int s;
+ for (s = 1; s < 64; s <<= 1)
+ w = (w >> s) + w;
+\end{alltt}
+
+
+\subsection{Hamming distance}
+\label{sec:Hamming distance}
+
+A simple way to compute the Hamming distance,
+the number of differing bits, between two number
+is with the function
+
+\begin{alltt}
+ size_t
+ hammdist(z_t a, z_t b)
+ \{
+ size_t ret;
+ z_t t;
+ zinit(t);
+ zxor(t, a, b);
+ ret = popcount(t);
+ zfree(t);
+ return ret;
+ \}
+\end{alltt}
+
+\noindent
+The performance of this function could
+be improve by comparing character by
+character manually with using {\tt zxor}.
+
+
+\subsection{Character retrieval}
+\label{sec:Character retrieval}
+
+\begin{alltt}
+ uint64_t
+ getu(z_t a)
+ \{
+ return zzero(a) ? 0 : a->chars[0];
+ \}
+\end{alltt}
(DIR) diff --git a/doc/number-theory.tex b/doc/number-theory.tex
@@ -0,0 +1,35 @@
+\chapter{Number theory}
+\label{chap:Number theory}
+
+TODO
+
+\vspace{1cm}
+\minitoc
+
+
+\newpage
+\section{Odd or even}
+\label{sec:Odd or even}
+
+TODO % zodd zeven zodd_nonzero zeven_nonzero
+
+
+\newpage
+\section{Signum}
+\label{sec:Signum}
+
+TODO % zsignum zzero
+
+
+\newpage
+\section{Greatest common divisor}
+\label{sec:Greatest common divisor}
+
+TODO % zgcd
+
+
+\newpage
+\section{Primality test}
+\label{sec:Primality test}
+
+TODO % zptest
(DIR) diff --git a/doc/random-numbers.tex b/doc/random-numbers.tex
@@ -0,0 +1,28 @@
+\chapter{Random numbers}
+\label{chap:Random numbers}
+
+TODO
+
+\vspace{1cm}
+\minitoc
+
+
+\newpage
+\section{Generation}
+\label{sec:Generation}
+
+TODO
+
+
+\newpage
+\section{Devices}
+\label{sec:Devices}
+
+TODO
+
+
+\newpage
+\section{Distributions}
+\label{sec:Distributions}
+
+TODO
(DIR) diff --git a/src/zmodmul.c b/src/zmodmul.c
@@ -6,6 +6,7 @@ void
zmodmul(z_t a, z_t b, z_t c, z_t d)
{
/* TODO Montgomery modular multiplication */
+ /* TODO Kochanski multiplication */
if (unlikely(a == d)) {
zset(libzahl_tmp_modmul, d);
zmul(a, b, c);
(DIR) diff --git a/src/zmodpow.c b/src/zmodpow.c
@@ -12,6 +12,8 @@ zmodpow(z_t a, z_t b, z_t c, z_t d)
size_t i, j, n, bits;
zahl_char_t x;
+ /* TODO use zmodpowu when possible */
+
if (unlikely(zsignum(c) <= 0)) {
if (zzero(c)) {
if (check(zzero(b)))
(DIR) diff --git a/src/zmodpowu.c b/src/zmodpowu.c
@@ -25,10 +25,11 @@ zmodpowu(z_t a, z_t b, unsigned long long int c, z_t d)
zmod(tb, b, d);
zset(td, d);
- zsetu(a, 1);
if (c & 1)
- zmodmul(a, a, tb, td);
+ zset(a, tb);
+ else
+ zsetu(a, 1);
while (c >>= 1) {
zmodsqr(tb, tb, td);
if (c & 1)
(DIR) diff --git a/src/zpow.c b/src/zpow.c
@@ -14,6 +14,8 @@ zpow(z_t a, z_t b, z_t c)
* 7↑19 = 7↑10011₂ = 7↑2⁰ ⋅ 7↑2¹ ⋅ 7↑2⁴ where a↑2↑(n + 1) = (a↑2↑n)².
*/
+ /* TODO use zpowu when possible */
+
size_t i, j, n, bits;
zahl_char_t x;
int neg;
(DIR) diff --git a/src/zpowu.c b/src/zpowu.c
@@ -21,10 +21,11 @@ zpowu(z_t a, z_t b, unsigned long long int c)
neg = znegative(b) && (c & 1);
zabs(tb, b);
- zsetu(a, 1);
if (c & 1)
- zmul_ll(a, a, tb);
+ zset(a, tb);
+ else
+ zsetu(a, 1);
while (c >>= 1) {
zsqr_ll(tb, tb);
if (c & 1)