Provided by: libbobcat-dev_5.09.01-2build1_amd64 
NAME
FBB::BigInt - Arithmetic on Integers of Unlimited Size
SYNOPSIS
#include <bobcat/bigint>
Linking option: -lbobcat -lcrypto
DESCRIPTION
This class is defined as a wrapper class around the openSSL BN series of functions,
offering members to perform arithmetic on integral values of unlimited sizes. Members are
offered to generate primes and to perform all kinds of common arithmetic operations on
BigInt objects. Also, conversions to characters and standard numerical value types are
offered.
Below, the phrase the object may also refer to the object’s value. The context in which
this occurs will make clear that the object’s value rather than the object as-is is
referred to.
Various constructors accept BIGNUM arguments. Type BIGNUM is the type containing an
integer of unlimited precision as defined by OpenSSL. BIGNUM’s definition is
typedef struct bignum_st BIGNUM;
struct bignum_st
{
BN_ULONG *d; // Pointer to an array of ’BN_BITS2’ bit chunks.
int top; // Index of last used d +1.
// The next are internal book keeping for bn_expand.
int dmax; // Size of the d array.
int neg; // one if the number is negative
int flags;
};
Signs of BigInt are handled in a special way. Whether a BigInt is negative or positive is
determined by its sign-flag, and not by a sign bit as is the case with int typed values.
Since BigInt values have unlimited precision shifting values to the left won’t change
their signs.
Operators return either a reference to the current (modified) object or return a BigInt
object containing the computed value. The rule followed here was to implement the
operators analogously to the way the operators work on int type values and variables.
E.g., operator+() returns a BigInt value whereas operator+=() returns a BigInt &
reference.
All members modifying their objects return a reference to the current (modified) object.
All members not modifying the current object return a BigInt object. If both members
exists performing the same functionality the name of the member returning a BigInt object
ends in a c (const) (e.g., addMod and addModc).
Almost all operators, members and constructors (except for the default constructor) throw
Exception exceptions on failure.
INHERITS FROM
-
TYPE
The class BigInt defines the type Word, which is equal to the type BN_ULONG used by
OpenSSL to store integral values of unlimited precision. A Word is an unsigned long, which
is, depending on the architecture, usually 64 or 32 bits long.
ENUMERATIONS
Msb
This (most significant bit) enumeration is used when generating a cryptographically strong
random number. Its values are:
o MSB_UNKNOWN:
The most significant bit may be 0 or 1.
o MSB_IS_ONE:
The most significant bit is guaranteed to be 1.
o TOP_TWO_BITS_ONE:
The two most significant bits are guaranteed to be 1, resulting in a product of two
values each containing nBits having 2 * nBits bits.
Lsb
This (least significant bit) enumeration is used when generating random numbers, ensuring
that the resulting value is either odd or even.
o EVEN:
The random value will be an even value;
o ODD:
The random value will be an odd value.
CONSTRUCTORS
o BigInt():
The default constructor initializes a BigInt value to 0.
o explicit BigInt(BIGNUM const &value):
This constructor initializes a BigInt from a const BIGNUM.
o explicit BigInt(BIGNUM const *value):
This constructor initializes a BigInt from a pointer to a const BIGNUM.
o explicit BigInt(BIGNUM *value):
This constructor initializes a BigInt from a pointer to a BIGNUM (the BIGNUM value
pointed to by value is not mondified by the constructor. This constructor is a
mere wrapper around the previous constructor).
o BigInt(Type value):
This constructor is defined as a member template. Any type that can be converted
using a static cast to an unsigned long can be used as argument to this
constructor. Promotion is allowed, so in many situations where BigInts are expected
a plain numerical value can be used as well.
o BigInt(char const *bigEndian, size_t length, bool negative = false):
This constructor initializes a BigInt from length big-endian encoded bytes stored
in bigEndian. This constructor interprets the char values pointed at by bigEndian
as unsigned values. Use this constructor to reconstruct a BigInt object from the
data made available by the bigEndian member. If the number represents a negative
value, then provide a third argument true.
o explicit BigInt(std::string const &bigEndian, bool negative = false):
This constructor initializes a BigInt from the bytes stored in bigEndian, which
must be big-endian encoded. This constructor interprets the char values stored in
bigEndian as unsigned values. If the number that is stored in bigEndian represents
a negative value, then provide a second argument true.
Copy and move constructors (and assignment operators) are available.
MEMBER FUNCTIONS
o BigInt &addMod(BigInt const &rhs, BigInt const &mod) :
Rhs is added (modulo mod) to the current object.
o BigInt addModc(BigInt const &rhs, BigInt const &mod) :
The sum (modulo mod) of the current object and rhs is returned.
o BigInt::Word at(size_t index) const:
Returns the Word at index. E.g., on a 32 bit architecture, if the BigInt value
equals 2, then at(0) returns 0, and at(1) returns 2. If index equals or exceeds the
value returned by nWords an FBB::Exception is thrown.
o BIGNUM const &bignum() const:
A reference to the BIGNUM value maintained by the current BigInt object is
returned.
o char *bigEndian() const:
The value represented by the current object is stored in a series of char typed
values in big-endian order. If a value consists of 5 chars the eight most
significant bits will be stored in the char having index value 0, the eight least
significant bits will be stored in the char having index value 4. When needed
simply swap char[i] with char[j] (i = 0 .. nBytes/2, j = nBytes-1 .. nBytes/2) to
convert to little-endian order. The return value consists of a series of
sizeInBytes() (see below) dynamically allocated char values. The caller of
bigEndian owns the allocated memory and should eventually delete it again using
delete[]. Note that the current object’s sign cannot be inferred from the return
value.
o BigInt &clearBit(size_t index):
The current object’s bit at index position index is cleared.
o BigInt clearBit(size_t index) const:
A copy of the current object having its bit at index position index cleared.
o BigInt &div(BigInt *remainder, BigInt const &rhs):
The current object is divided by rhs. The division’s remainder is returned in
*remainder.
o BigInt divc(BigInt *remainder, BigInt const &rhs) const:
The quotient of the current object and rhs is returned. The division’s remainder is
returned in *remainder.
o int compare(BigInt const &rsh) const:
Using signed values, if the current object is smaller than rhs -1 is returned; if
they are equal 0 is returned; if the current object is larger than ths 1 is
returned (see also uCompare).
o BigInt &exp(BigInt const &exponent):
The current object is raised to the power exponent.
o BigInt expc(BigInt const &exponent) const:
The current object raised to the power exponent is returned.
o BigInt &expMod(BigInt const &exponent, BigInt const &mod):
The current object is raised to the power exponent modulo mod.
o BigInt expModc(BigInt const &exponent, BigInt const &mod) const:
The current object raised to the power exponent modulo mod is returned.
o BigInt &gcd(BigInt const &rhs):
The greatest common divisor (gcd) of the current object and rhs is assigned to the
current object. To compute the least common multiple (lcm) the following
relationship can be used:
lcm(a, b) = a * b / a.gcd(b)
o BigInt gcdc(BigInt const &rhs) const:
The greatest common divisor (gcd) of the current object and rhs is returned. To
compute the least common multiple (lcm) the following relationship can be used:
lcm(a, b) = a * b / a.gcd(b)
o bool hasBit(size_t index):
True is returned if the bit at index position index has been set, false otherwise.
o BigInt &inverseMod(BigInt const &mod):
The inverse of the current object modulo mod is assigned to the current object.
This is the value ret for which the following expression holds true:
(*this * ret) % mod = 1
o BigInt inverseModc(BigInt const &mod) const:
This inverse of the current object modulo mod is returned.
o bool isNegative() const:
Returns true if the current object contains a negative value.
o bool isOdd() const:
Returns true if the current object is an odd value.
o bool isOne() const:
Returns true if the current object equals one (1).
o BigInt &isqrt():
The current object’s integer square root value is assigned to the current object.
The integer square root of a value x is the biggest integral value whose square
does not exceed x. E.g., isqrt(17) == 4. An Exception exception is thrown if the
current object’s value is smaller than one.
o BigInt isqrtc() const:
The integer square root of the current object is returned. An Exception exception
is thrown if the current object’s value is smaller than one.
o bool isZero() const:
Returns true if the current object equals zero (0).
o BigInt &lshift():
The current object’s bits are shifted one bit to the left. The object’s sign
remains unaltered.
o BigInt lshiftc():
The current object’s bits shifted one bit to the left are returned. The object’s
sign will be equal to the current object’s sign.
o BigInt &lshift(size_t nBits):
The current object’s bits are shifted nBits to the left. The object’s sign remains
unaltered.
o BigInt lshiftc(size_t nBits) const:
The current object’s bits shifted nBits bit to the left are returned. The object’s
sign will be equal to the current object’s sign.
o BigInt &maskBits(size_t lowerNBits):
The current object’s lowerNBits lower bits are kept, its higher order bits are
cleared. The object’s sign is not affected.
o BigInt maskBitsc(size_t lowerNBits) const:
A copy of the current object is returned having all but its lowerNBits lower bits
cleared. The sign of the returned object will be equal to the current object’s
sign.
o size_t maxWordIndex() const:
Returns the maximum Word-index that can be used with the at and setWord members for
the current BigInt value.
o BigInt &mulMod(BigInt const &rhs, BigInt const &mod):
The current object is multiplied (modulo mod) by rhs.
o BigInt mulModc(BigInt const &rhs, BigInt const &mod) const:
The current object multiplied (modulo mod) by rhs is returned.
o BigInt &negate():
The current object’s value is negated (i.e., the value changes its sign).
o BigInt negatec() const:
The negated value of the current object is returned.
o size_t nWords() const:
The number of `words’ required to store the BigInt value is returned. Note that the
returned value depends on the architecture’s number of bytes per word. For 32-bit
architectures there are four bytes per word, for 64-bit architectures eight bytes
per word.
o BigInt &rshift():
The current object’s bits are shifted one bit to the right. The object’s sign
remains unaltered.
o BigInt rshiftc():
The current object’s bits shifted one bit to the right are returned. The object’s
sign will be equal to the current object’s sign.
o BigInt &rshift(size_t nBits):
The current object’s bits are shifted nBits to the right. The object’s sign remains
unaltered.
o BigInt rshiftc(size_t nBits) const:
The current object’s bits shifted nBits bit to the right are returned. The object’s
sign will be equal to the current object’s sign.
o BigInt &setBit(size_t index):
The bit at index position index is set.
o BigInt setBitc(size_t index) const:
A copy of the current object is returned having its bit at index position index
set.
o BigInt &setBit(size_t index, bool value):
The bit at index position index is set to value.
o BigInt setBitc(size_t index, bool value) const:
A copy of the current object is returned having its bit at index position index set
to value.
o BigInt &setNegative(bool negative):
The current object’s sign will be set to negative if the function’s argument is
true, it will be set to positive if the function’s argument is false.
o BigInt setNegativec(bool negative) const:
A copy of the current object is return having a negative sign if the function’s
argument is true and a positive sign if the function’s argument is false.
o void setWord(size_t index, BigInt::Word value):
Assigns value to the Word at index. E.g., on a 32 bit architecture, if the BigInt
value equals 2, then after setWord(1, 1) the value has become 2. If index exceeds
the value returned by nWords an FBB::Exception is thrown.
o size_t size() const:
The number of significant bits required to store the current BIGNUM value is
returned.
o size_t sizeInBytes() const:
The number of bytes required to store the current BIGNUM value is returned (returns
the same value as the size member does).
o size_t constexpr sizeOfWord() const:
BigInt values are stored in units of `words’, which are unsigned long values. These
values may consist of, e.g., 32 or 64 bits. The number of bytes occupied by a
`word’ is returned: 4 for a 32 bit value, 8 for a 64 bit value, and possibly other
values, depending on specific architecture peculiarities. The value returned by
this member, therefore, is architecture dependent.
o BigInt &sqr():
The current object’s value is squared.
o BigInt sqrc() const:
The square of the current object is returned.
o BigInt &sqrMod(BigInt const &mod) const:
The current object’s value is squared modulo mod.
o BigInt sqrModc(BigInt const &mod) const:
The square (modulo mod) of the current object is returned.
o BigInt &subMod(BigInt const &rhs, BigInt const &mod):
Rhs is subtracted modulo mod from the current object.
o BigInt subModc(BigInt const &rhs, BigInt const &mod) const:
The difference (modulo mod) of the current object and rhs is returned.
o void swap(BigInt &other):
The current object swaps its value with other.
o BigInt &tildeBits():
All the bits in the bytes of the current object and the sign of the current object
are toggled. So, after
Bigint b(5);
b.tildeBits();
b contains the value -250. Also see the discussion with operator~() below.
o BigInt tildeBitsc() const:
A copy of the current object whose bits are toggled is returned.
o BigInt &tildeInt():
The `tilde’ operation is performed on the current object using the standard int
semantics. E.g., ~5 results in -6. Also see the discussion with operator~() below.
o BigInt tildeIntc() const:
A copy of the current object is returned to which the `tilde’ operation has been
performed using the standard int semantics.
o unsigned long ulong() const:
The absolute value stored in the current object is returned as an unsigned long. If
it cannot be represented by an unsigned long it returns 0xffffffffL.
o int uCompare(BigInt const &rsh) const:
Using absolute values, if the current object is smaller than rhs -1 is returned; if
they are equal 0 is returned; if the current object is larger than ths 1 is
returned (see also uCompare).
OVERLOADED OPERATORS
Except for some operators all operators perform their intuitive operations. Where that
isn’t completely true an explanatory remark is provided. E.g., operator*() multiplies two
BigInts, possibly promoting one of the operands; operator*=() multiplies the lhs by the
rhs BigInt, possibly promoting the rhs operand.
Here are the available operators:
Unary operators:
o bool operator bool() const:
Returns true if the BigInt value is unequal zero, otherwise false is returned.
o BigInt &operator++():
o BigInt operator++(int):
o BigInt &operator--():
o BigInt operator--(int):
o BigInt operator-():
o int operator[](size_t idx) const:
With BigInt objects it returns the bit-value of the object’s idxth bit as the value
0 or 1.
o BigInt::Bit operator[](size_t idx):
With non-const BigInt objects it returns a reference to the bit-value of the
object’s idxth bit. When used as lvalue assigning a 0 or non-zero value to the
operator’s return value will either clear or set the bit. Likewise, the following
arithmetic assignment operators may be used: binary or (|=), binary and (&=) or
binary xor (^=). When used as rvalue the value of the object’s idxth bit is
returned as a bool value. When inseerted into a std::ostream the bit’s value is
displayed as 0 or 1.
o BigInt operator~():
This operator is not implemented as it cannot be implemented so that it matches the
actions of this operator when applied to int type values.
When used on int values this operator toggles all the int’s bits. E.g., ~5
represents -6, and ~-6 again equals five. The -6 is the result of the sign bit of
int values. The obvious implementation of BigInt::operator~() is to toggle all the
value’s bits and to toggle its sign bit. For 5 this would result in -250: 5, being
101 (binary), fits in one byte, so ~5 becomes 11111010 (binary), which is 250. Its
sign must be reversed as well, so it becomes -250. This clearly differs from the
value represented by the int constant ~5: when constructing BigInt(~5), the value
-6 is obtained.
It is possible to change the implementation. E.g., after
Bigint b(5);
b = ~b;
~b could be implemented so that it results in the value -6. But this too leads to
unexpected results. While 5 & ~5 == 0, this would no longer hold true for BigInt
objects: Assuming b contains 5 then b & ~b would expand to (binary) 101 &
(negative)110 which equals (binary) 100.
Since either implementation produces unexpected results BigInt::operator~() was not
implemented. Instead two members are offered: tildeBits(), toggling all the bits of
all the BigInt bytes and toggling its sign (so
Bigint b(5);
b.tildeBits();
changes b’s value into -250), and tildeInt() changing the object’s value into the
value that would have been obtained if a BigInt was a mere int (so
Bigint b(5);
b.tildeInt();
changes b’s value into -6).
Binary operators:
o BigInt operator*(BigInt const &lhs, BigInt const &rhs):
o BigInt operator/(BigInt const &lhs, BigInt const &rhs):
This operator returns the quotient of the lhs object divided by the rhs object. The
remainder is lost (The member div performs the division and makes the remainder
available as well).
o BigInt operator%(BigInt const &lhs, BigInt const &rhs):
o BigInt operator+(BigInt const &lhs, BigInt const &rhs):
o BigInt operator-(BigInt const &lhs, BigInt const &rhs):
o BigInt operator<<(BigInt const &lhs, size_t nBits):
See also the lshift members. If lhs is positive,
o BigInt operator>>=(BigInt const &lhs, size_t nBits):
See also the rshift members.
o BigInt operator&(BigInt const &lhs, BigInt const &rhs):
This operator returns a BigInt value consisting of the bit_and-ed bits and sign
flags of lhs and rhs operands. Consequently, if one operand is positive, the
resulting value will be positive.
o BigInt operator|(BigInt const &lhs, BigInt const &rhs):
This operator returns a BigInt value consisting of the bit_or-ed bits and sign
flags of lhs and rhs operands. Consequently, if either operand is negative, the
result will be negative.
o BigInt operator^(BigInt const &lhs, BigInt const &rhs):
This operator returns a BigInt value consisting of the bit_xor-ed bits and sign
flags of lhs and rhs operands. Consequently, if exactly one operand is negative,
the result will be negative.
(Arithmetic) assignment operator(s):
o BigInt &operator*=(BigInt const &rhs):
o BigInt &operator/=(BigInt const &rhs):
This operator assigns the result of the (integer) division of the current BigInt
object by ths to the current object. The remainder is lost. The member div divides
and makes the remainder available as well.
o BigInt &operator%=(BigInt const &rhs):
o BigInt &operator+=(BigInt const &rhs):
o BigInt &operator-=(BigInt const &rhs):
o BigInt &operator<<=(size_t nBits):
See also the lshift members.
o BigInt &operator>>=(size_t nBits):
See also the rshift members.
o BigInt &operator&=(BigInt const &rhs):
This operator bit_ands the bits and sign flags of the current object and the rhs
operand.
o BigInt &operator|=(BigInt const &rhs):
This operator bit_ors the bits and sign flags of the current object and the rhs
operand.
o BigInt &operator^=(BigInt const &rhs):
This operator bit_xors the bits and sign flags of the current object and the rhs
operand.
STATIC MEMBERS
All members returning a BigInt computed from a set of arguments and not requiring an
existing BigInt object are defined as static members. The first diophantus member,
returning a long long value, also is a static member.
o long long diophantus(long long *factor1, long long *factor2, long long value1, long
long value2):
The integral solution of factor1 * value1 + factor2 * value2 = gcd is computed. The
function returns the greatest common divisor (gcd) of value1 and value2, and
returns their multiplication factors in, respectively, *factor1 and *factor2. The
solution is not unique: another solution is obtained by adding k * value2 to
factor1 and subtracting k * value1 from factor2. For values exceeding
std::numeric_limits<long, long>::max() the next member can be used.
o BigInt diophantus(BigInt *factor1, BigInt *factor2, BigInt const &value1, BigInt
const &value2):
The integral solution of factor1 * value1 + factor2 * value2 = gcd is computed. The
function returns the greatest common divisor (gcd) of value1 and value2, and
returns their multiplication factors in, respectively, *factor1 and *factor2. The
solution is not unique: another solution is obtained by adding k * value2 to
factor1 and subtracting k * value1 from factor2.
o BigInt fromText(std::string text, int mode = 0):
This member converts a textual representation of a number to a BigInt value.
Conversion continues until the end of text or until a character outside of an
expected range is encountered.
The expected range may be preset by specifying mode as ios::dec, ios::oct, or
ios::hex or (the default) the expected range is determined by fromText itself by
inspecting the characters in text.
By default if text contains hexadecimal characters then fromText assumes that the
number is represented as a hexadecimal value (e.g., "abc" is converted to the
(decimal) value 2748); if text starts with 0 and contains only characters in the
range 0 until (including) 7 then fromText assumes the number is represented as an
octal value (e.g., "01234" is converted to the (decimal) value 668). Otherwise a
decimal value is assumed.
If the text does not represent a valid numerical value (of the given extraction
mode) then a FBB::Exception exception is thrown (fromText: text does not represent
a BigInt value).
o BigInt rand(size_t size, Msb msb = MSB_IS_ONE, Lsb lsb = ODD):
This member returns a cryptographically strong pseudo-random number of size bits.
The most significant bit(s) can be controlled by msb (by default MSB_IS_ONE), the
least significant bit can be controlled by lsb (by default ODD). Before calling
this member the random number generator must have been seeded.
From the RAND_add(3ssl) man-page:
OpenSSL makes sure that the PRNG state is unique for each thread. On systems that
provide /dev/urandom, the randomness device is used to seed the PRNG transparently.
However, on all other systems, the application is responsible for seeding the PRNG
by calling RAND_add(3ssl), RAND_egd(3ssl), RAND_load_file(3ssl), or
RAND_seed(3ssl).
o BigInt randRange(BigInt const &max):
This member returns a cryptographically strong pseudo-random number in the range 0
<= number < max. Before calling this member the random number generator must have
been seeded (see also rand, described above).
o BigInt setBigEndian(std::string const &bytes):
The bytes.length() bytes of bytes are used to compute a BigInt object which is
returned by this function. The characters in bytes are interpreted as a series of
bytes in big-endian order. See also the member function bigEndian() above. The
returned BigInt has a positive value.
o BigInt prime(size_t nBits, BigInt const *mod = 0, BigInt const *rem = 0, PrimeType
primeType = ANY):
This member returns a prime number of bBits bits. If both mod and rem are non-zero,
the condition prime % mod == rem. (E.g., use prime % mod == 1 in order to suit a
given generator). The parameter primeType can be ANY, (prime - 1) / 2 may or may
not be a prime. If it is SAFE then (prime - 1) / 2 will be a (so-called safe)
prime.
o BigInt pseudoRand(size_t size, Msb msb = MSB_IS_ONE, Lsb lsb = ODD):
This member returns a potentially predictable pseudo-random number of size bits.
The most significant bit(s) can be controlled by msb (by default MSB_IS_ONE), the
least significant bit can be controlled by lsb (by default ODD). It can be used for
non-cryptographic purposes and for certain purposes in cryptographic protocols, but
usually not for key generation.
o BigInt pseudoRandRange(BigInt const &max):
This member returns a potentially predictable pseudo-random number in the range 0
<= number < max.
FREE FUNCTIONS IN THE FBB NAMESPACE
o std::ostream &operator<<(ostream &out, BigInt const &value):
Inserts value into the provided ostream. If the hex manipulator has been inserted
into the stream before inserting the BigInt value the value will be displayed as a
hexadecimal value (without a leading 0x); if the oct manipulator has been inserted
the value will be represented as an octal value (starting with a 0). The value will
be displayed as a decimal value if the dec manipulator is active. If the BigInt
value is negative its value will be preceded by a minus character.
o std::istream &operator>>(istream &in, BigInt &value):
Extracts value from the provided istream. Depending on the currently set extraction
mode (dec, oct, or hex) the matching set of characters will be extracted from in
and converted to a number which is stored in value. Extraction stops at EOF or at
the first character outside of the range of characters matching the extraction
mode. if no numerical characters were extracted the stream’s failbit is set. The
extracted value may be preceded by a minus character, resulting in an extracted
negative value.
EXAMPLE
#include <iostream>
#include <bobcat/bigint>
using namespace std;
using namespace FBB;
int main()
{
BigInt value(BigInt::prime(100));
BigInt mod(BigInt::rand(50));
BigInt inverse(value.inverseModc(mod));
cout << ’(’ << value << " * " << inverse << ") % " << mod << " = " <<
( value * inverse ) % mod << endl;
}
FILES
bobcat/bigint - defines the class interface
SEE ALSO
bobcat(7), diffiehellman(3bobcat), RAND_add(3ssl), RAND_egd(3ssl), RAND_load_file(3ssl),
RAND_seed(3).
For BIGNUM: https://www.openssl.org/docs/man1.0.2/man3/bn_sub_words.html
BUGS
Sep/Oct 2013: due to a change in library handling by the linker (cf.
http://fedoraproject.org/wiki/UnderstandingDSOLinkChange and
https://wiki.debian.org/ToolChain/DSOLinking) libraries that are indirectly required are
no longer automatically linked to your program. With BigInt this is libcrypto, which
requires programs to link to both bobcat and crypto.
BOBCAT PROJECT FILES
o https://fbb-git.gitlab.io/bobcat/: gitlab project page;
o bobcat_5.09.01-x.dsc: detached signature;
o bobcat_5.09.01-x.tar.gz: source archive;
o bobcat_5.09.01-x_i386.changes: change log;
o libbobcat1_5.09.01-x_*.deb: debian package containing the libraries;
o libbobcat1-dev_5.09.01-x_*.deb: debian package containing the libraries, headers
and manual pages;
BOBCAT
Bobcat is an acronym of `Brokken’s Own Base Classes And Templates’.
COPYRIGHT
This is free software, distributed under the terms of the GNU General Public License
(GPL).
AUTHOR
Frank B. Brokken (f.b.brokken@rug.nl).