FBB::BigInt - Arithmetic on Integers of Unlimited Size
#include <bobcat/bigint>
Linking option: -lbobcat -lcrypto
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.
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.
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.
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.
- 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.
The standard copy constructor is available, the move constructor
is not available.
- 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 memeber 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).
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):
- 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.
Note that the move operator is not available
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.
- 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.
#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;
}
bobcat/bigint - defines the class interface
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.
- o
- bobcat_4.08.06-x.dsc: detached signature;
- o
- bobcat_4.08.06-x.tar.gz: source archive;
- o
- bobcat_4.08.06-x_i386.changes: change log;
- o
- libbobcat1_4.08.06-x_*.deb: debian package holding the
libraries;
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- libbobcat1-dev_4.08.06-x_*.deb: debian package holding the
libraries, headers and manual pages;
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- http://sourceforge.net/projects/bobcat: public archive location;
Bobcat is an acronym of `Brokken’s Own Base Classes And
Templates’.
This is free software, distributed under the terms of the GNU
General Public License (GPL).
Frank B. Brokken (f.b.brokken@rug.nl).