Proposal for Unbounded-Precision Integer Types

What’s New in this Revision

The discussion in Bristol resulted in several suggested changes. I have incorporated those comments as new issues in Open issues; they’re marked [new in N3965].

• added member functions integer::mod, integer::mulmod, and integer::powmod; added free functions mod, mulmod, powmod, gcd, and lcm for objects of type integer;
• added language to allow implementations to use expression templates;
• added noexcept qualifiers to all appropriate functions; in particular, all constructors that take integral types are marked noexcept, which implies that implementations must implement a small-object optimization for integral types.
• removed bits::highest_bit and bits::lowest_bit; the first is too hard to specify in a meaningful way for an unbounded set, and the second seems vestigial without the first.
• replaced all mentions of size_type with size_t.

Overview

Programmers sometimes need to manipulate integer values that are too large to repesent with C++’s standard integer types. Doing a Google search for terms that describe large integers produces many hits for libraries that handle large integers. These libraries vary in quality, from hacks by beginners to sophisticated, professional implementations. Also, Java has unbounded precision integers as part of its standard class library.

One important use for unbounded-precision integers is cryptography. Cryptographic applications typically manipulate integer values of several hundred digits. If the C++ standard library provides facilities for such values it will make cryptographic applications easier to write and to port.

There have been two Committee papers proposing unbounded-precision integer libraries for C++: N1718 (2004) and N2143 (2007), both by M.J. Kronenburg. Nothing was done with these papers, in part because there was too much else going on in the Library Working Group at the time. Now that the Committee is looking to greatly expand the scope of the standard library, it’s time to reconsider unbounded-precision integers.

Design considerations

An unbounded-precision integer type is an integer type that does not impose pre-defined limits on the precision of the values that it can represent. Of course, in practice, unbounded precision can’t be achieved; sooner or later the system runs out of resources. So unbounded in this context means bounded only by the availability of system resources; there is no hard-coded limit to the number of digits in the value that an unbounded-precision integer type an represent.

Unbounded-precision integer types should interoperate with C++’s built-in integer types. Applying arithmetic operations to a mix of standard integer types and unbounded-precision integer types should just work. And to the extent possible, operations that can be applied to standard integer types should also be applicable, using the same syntax, to unbounded-precision integer types.

Unbounded-precision integer types should also provide operations that facilitate their use in areas that demand such types. Since there is a potentially unbounded list of operations that could be useful in applications that need unbounded-precision integer types, it is not practical to provide every useful operation. This proposal presents a small set of required operations and provides a facility for users to write their own extensions.

Design overview

This paper proposes two unbounded-precision integer types. The type integer represents signed integer values. The type bits represents an unbounded set of bit values.

To support interoperability, objects of either type can be constructed from values of any of the standard integer types. So code like this just works:

    integer i = 30000;
    integer j = 1000 * i;

    bits b = 0xFF;
    bits c = b & 0xAA;

Converting a negative number to an object of type bits sets the number to the complement of this initializer, so this code just works:

    bits b = -3; // sets b to ...11111100

This is easily implemented by storing a finite set of 1 bits (in this case, 0x03) and using a flag to indicate whether the actual value is represented by that set of bits or by its complement.

As currently specified, neither integer nor bits provides overloaded operators that take standard integer types. When an operation is applied to a standard integer type and an integer object or a bits object, the standard integer operand is converted to the appropriate unbounded-precision integer type and the operation is applied to the result. It is assumed that implementations will make this kind of mixed operation efficient by implementing a small-integer optimization, storing small values directly in the integer or bits object, and using heap storage when needed for larger values. This greatly simplifies the interface specification.

Objects of these types can be constructed from string representations (with the usual range of possible bases) and from an initializer list that holds unsigned 32-bit values.

Objects of type integer can also be constructed from values of floating-point types.

Values of type integer and of type bits can be freely inter-converted.

Bit manipulations on bits objects treat the value as having an unbounded number of bits above the highest bit stored in the object. As a result, the usual bit operations, &, |, and ^, can be applied to values of different sizes. Further, std::seminumeric::bits can be used as a replacement for std::bitset when limiting the object to a fixed number of bits is undesirable.

Issues

Open Issues

• 1. Allocators -- unbounded-precision integer types need to manage memory on the free store to hold their data. This suggests that users should be able to specify an allocator. But templatizing these types on an allocator would require that all of the arithmetic operations provide overloads for all of the standard integer types, because function template instantiation requires exact type matches, without conversions.
• 5. [new in N3417] Rvalue overloads for arithmetic operations (Joe Gottman) -- many of the operations on the unbounded-precision types can be made more efficient by providing overloaded versions that take rvalue references. For example:

  integer operator+(integer&& lhs, const integer& rhs) {
      return std::move(lsh += rhs);
  }

• 7. [new in N3417] Support user-defined literals (Alisdair Meredith, Marc Glisse) -- obviously useful, but users might expect them to be constexpr, which they cannot, in general, be, because they need to allocate memory.
• 9. [new in N3417] Add constexpr specifications as appropriate (Alisdair Meredith) -- although most functions can allocate memory, there are a few that could be marked constexpr. It’s not clear how useful this would be.
• 12. [new in N3417] Converting to a string could use a mechanism for providing an allocator (Alisdair Meredith) -- for example,

  template<class charT, class Traits, class Alloc>
  void bits::load(basic_string<charT, Traits, Alloc>&);

• 14. [new in N3417] powmod should be specified to run in constant time with identical cache access patterns for arguments of the same size (Jack Lloyd) -- this is important for cryptographic purposes, to avoid side-channel attacks.
• 15. [new in N3417] Need functions to interconvert with byte arrays (Jack Lloyd) -- quite a common need in cryptographic operations.
• 18. [new in N3417] Rounding of divisions and right shifts needs to be specified. (Marc Glisse) -- even if it’s obvious. Also, does mod return negative values?
• 24. [new in N3965] Provide interoperability with fixed point.
• 25. [new in N3965] What happens when integer_data_proxy::operator[] is called with an index that is out of bounds?

Issues Closed in N3965

• 4. [resolved in N3965] Policy for lossy conversions -- converting an object of type bits or integer to an integral type that cannot represent its value doesn’t have any specified semantics in this paper. Should this throw an exception? Resolution: conversions of values that cannot be represented in the target type throw an exception of type std::range_error.
• 6. [resolved in N3965][new in N3417] Functions for controlling when and how allocations occur (Joe Gottman) -- just as with std::string, many operations can be made faster by pre-allocating enough memory to hold the expected result. In some cases the desired capacity is best expressed in bits and in some cases in decimal digits, so the suggestion is to offer both. Resolution: integer capacities are expressed in decimal digits (but integer_data_proxy capacity is in number of internal data elements; does this make sense?); bits capacities are expressed in number of bits.

  size_t capacity_in_bits() const;   // The number of bits the object can hold without reallocation.
  size_t capacity_in_digits() const; // The largest number n such that the object can hold n decimal
                                     // digits without reallocation.
  void reserve_bits(size_t n);       // Postcondition: capacity_in_bits() >= n.
  void reserve_digits(size_t n);     // Postcondition: capacity_in_digits() >= n.
  void shrink_to_fit()();            // A non-binding request to reduce memory usage.

• 8. [resolved in N3965] [new in N3417] Add noexcept specifications as appropriate (Alisdair Meredith) -- there are several that are obvious. In addition, if we require the small-object optimization, some or all of the constructors that take integral types can be noexcept. Resolution: added noexcept; constructors that take integral types are marked noexcept, implying a small-object optimization.
• 10. [resolved in N3965] [new in N3417] Why do the bitshift operators take an int argument? (Alisdair Meredith, Marc Glisse) -- this may be limiting, for example, on a 64-bit OS with a 32-bit int type. size_t or ptrdiff_t may be a better choice. Resolution: shift operators take an argument of type size_t.
• 22. [resolved in N3965] [new in N3965] Restore mod, mulmod, powmod, gcd, lcm. Resolution: done.
• 23. [resolved in N3965] [new in N3965] Ensure that requirements do not preclude implementation with expression templates. Resolution: done, as suggested in the Bristol minutes, except that the second sentence is not included: the class integer in not a template.

Issues Closed Prior to N3965

• 2. [resolved in N3542] Policy for out-of-bounds subtraction -- subtracting an unsigned_integer value from a smaller value doesn’t have any specified semantics in this paper. This should probably throw an exception. Resolution: removed unsigned_integer. The replacement type bits does not provide arithmetic operations.
• 3. [resolved in N3542] Policy for out-of-bounds constructor arguments -- constructing a bits object from a negative value doesn’t have any specified semantics in this paper. This should probably throw an exception. Resolution: negative values produce complemented bit sets.
• 11. [resolved in N3542] [new in N3417] Why just one string type for string conversions? (Alisdair Meredith) -- there are four standard basic_string typedefs. Resolution: string operations are now fully templated.
• 13. [resolved in N3542] [new in N3417] Consider extending interface to more fully support dynamic bitsets (Alisdair Meredith) -- or maybe that should be a separate class, with shared implementation. Resolution: changed unsigned_integer to bits with same interface as std::bitset.
• 16. [resolved in N3542] [new in N3417] Add a typedef that gives the underlying representation type and an accessor that provides a pointer to the internal representation. (Jack Lloyd) -- some common cryptographic algorithms can’t be implemented with the interface in the proposal; adding these accessors makes it possible to implement them. Resolution: added class integer_data_proxy.
• 17. [resolved in N3542] [new in N3417] Is the unsigned_integer type necessary? (Marc Glisse) -- if this is there to support bit manipulation, why not just specify those operations as undefined for negative numbers. Resolution: removed unsigned_integer and added bits, whose sole purpose is bit manipulation.
• 19. [resolved in N3542] [new in N3417] The constructor that takes a string should be explicit. (Marc Glisse) Resolution: done.
• 20. [resolved in N3542] [new in N3417] Consider making to_string() a non-member function. (Daniel Krügler) -- this would be a good match with the existing to_string functions. Resolution: done for integer; bits has it as a member to match std::bitset.
• 21. [resolved in N3542] [new in N3417] Consider removing explicit operator std::string() const. (Daniel Krügler, Jens Maurer) -- to_string() is sufficient. Resolution: done.

Header <seminumeric> synopsis

namespace std {
namespace experimental {
namespace seminumeric {

/* class integer */

    class integer;
    class integer_data_proxy;

    void swap(integer& lhs, integer& rhs) noexcept;

    // comparisons
    bool operator==(const integer& lhs, const integer& rhs) noexcept;
    bool operator!=(const integer& lhs, const integer& rhs) noexcept;
    bool operator<(const integer& lhs, const integer& rhs) noexcept;
    bool operator<=(const integer& lhs, const integer& rhs) noexcept;
    bool operator>(const integer& lhs, const integer& rhs) noexcept;
    bool operator>=(const integer& lhs, const integer& rhs) noexcept;

    // arithmetic operations
    integer operator+(const integer& lhs, const integer& rhs);
    integer operator-(const integer& lhs, const integer& rhs);
    integer operator*(const integer& lhs, const integer& rhs);
    integer operator/(const integer& lhs, const integer& rhs);
    integer operator%(const integer& lhs, const integer& rhs);

    std::pair<integer, integer> div(const integer& lhs, const integer& rhs);

    integer abs(const integer& val);

    integer operator<<(const integer& lhs, size_t rhs);
    integer operator>>(const integer& lhs, size_t rhs);

    // numeric operations
    integer sqr(const integer& val);
    integer sqrt(const integer& val);
    integer pow(const integer& val, const integer& exp);
    integer mod(const integer& lhs, const integer& rhs);
    integer mulmod(const integer& lhs, const integer& rhs, const integer& m);
    integer powmod(const integer& lhs, const integer& rhs, const integer& m);

    integer gcd(const integer& a, const integer& b);
    integer lcm(const integer& a, const integer& b);


    // conversions
    std::string to_string(const integer& val, int radix = 10);

    // I/O operations
    template <class CharT, class Traits>
        std::basic_ostream<CharT, Traits>& operator<<(
            std::basic_ostream<CharT, Traits>& str, const integer& val);
    template <class CharT, class Traits>
        std::basic_istream<CharT, Traits>& operator>>(
            std::basic_istream<CharT, Traits>& str, integer& val);

/* class bits */

    class bits;

    void swap(bits& lhs, bits& rhs) noexcept;

    // logical operations
    bits operator&(const bits& lhs, const bits& rhs);
    bits operator|(const bits& lhs, const bits& rhs);
    bits operator^(const bits& lhs, const bits& rhs);

    // I/O operations
    template <class CharT, class Traits>
        std::basic_ostream<CharT, Traits>& operator<<(
            std::basic_ostream<CharT, Traits>& str, const bits& val);
    template <class CharT, class Traits>
        std::basic_istream<CharT, Traits>& operator>>(
            std::basic_istream<CharT, Traits>& str, bits& val);

} /* namespace seminumeric */
} /* namespace experimental */

template <class Ty> class numeric_limits<Ty>;
template <> class numeric_limits<experimental::seminumeric::integer>;

template <class Ty> class hash<Ty>;
template <> class hash<experimental::seminumeric::integer>;
template <> class hash<experimental::seminumeric::bits>;

} /* namespace std */

Class integer

class integer {
public:

    // constructors
    integer() noexcept;

    integer(char val) noexcept;
    integer(signed char val) noexcept;
    integer(unsigned char val) noexcept;
    integer(short val) noexcept;
    integer(unsigned short val) noexcept;
    integer(int val) noexcept;
    integer(unsigned val) noexcept;
    integer(long val) noexcept;
    integer(unsigned long val) noexcept;
    integer(long long val) noexcept;
    integer(unsigned long long val) noexcept;

    integer(float val);
    integer(double val);
    integer(long double val);

    integer(std::initializer_list<uint_least32_t> init);

    template <class CharT, class Traits, class Alloc>
        explicit integer(const std::basic_string<CharT, Traits, Alloc>& str);

    explicit integer(const bits& rhs);
    explicit integer(bits&& rhs);

    integer(const integer& rhs);
    integer(integer&& rhs) noexcept;

    // assign and swap
    integer& operator=(const integer& rhs);
    integer& operator=(integer&& rhs);
    integer& operator=(const bits& rhs);
    integer& operator=(bits&& rhs);
    void swap(integer& rhs);

    // conversions
    explicit operator long long() const;
    explicit operator unsigned long long() const;
    explicit operator long double() const noexcept;
    explicit operator bool() const noexcept;

    // comparisons
    int compare(const integer& rhs) const noexcept;

    // arithmetic operations
    integer& operator+=(const integer& rhs);
    integer& operator-=(const integer& rhs);
    integer& operator*=(const integer& rhs);
    integer& operator/=(const integer& rhs);
    integer& operator%=(const integer& rhs);

    integer& operator++();
    integer operator++(int);
    integer& operator--();
    integer operator--(int);

    integer div(const integer& rhs);

    integer& abs() noexcept;
    integer& negate() noexcept;
    integer operator+() const;
    integer operator-() const;

    integer& operator<<=(size_t rhs);
    integer& operator>>=(size_t rhs);

    // numeric operations
    integer& sqr();
    integer& sqrt();
    integer& pow(const integer& exp);
    integer& mod(const integer& rhs);
    integer& mulmod(const integer& rhs, const integer& m);
    integer& powmod(const integer& exp, const integer& m);

    // observers
    bool is_odd() const noexcept;

    // accessors
    integer_data_proxy get_data_proxy();

    // capacity
    size_t size() const noexcept;
    size_t capacity() const noexcept;
    void reserve(size_t digits);
    void shrink_to_fit();
};

The class describes an object that manages an unbounded-precision signed integral type that can be used in most contexts where an int could be used.

Any function specified to return an object of type integer may return an object of another type, provided all the const member functions of the class integer are also applicable to that type.

integer abs(const integer& other);
integer& integer::abs() noexcept;

The first function returns an object that holds the absolute value of other. The second function sets the stored value of *this to its absolute value and returns *this.

size_t integer::capacity() const noexcept;

The member function returns the number of decimal digits that the object can represent without reallocating its internal storage.

int integer::compare(const integer& rhs) const noexcept;

The member function returns a value less than 0 if *this is less than rhs, 0 if *this is equal to rhs, and greater than 0 if *this is greater than rhs.

std::pair<integer, integer> div(const integer& lhs, const integer& rhs);
std::pair<integer, integer> integer::div(const integer& rhs) const;

The functions return an object that is an instantiation of std::pair; its first field holds the quotient, lhs / rhs or *this / rhs, and its second field holds the remainder, lhs % rhs or *this % rhs.

integer gcd(const integer& a, const integer& b);

The function return returns an object whose value is the greatest common denominator of a and b.

integer_data_proxy integer::get_data_proxy();

The member function returns an object of type integer_data_proxy that can be used to examine and modify the internal storage of *this. If an object of type integer_data_proxy that refers to *this exists at the time of a call to this function, the function throws an exception object of type std::runtime_error.

integer::integer() noexcept;

integer::integer(char val) noexcept;
integer::integer(signed char val) noexcept;
integer::integer(unsigned char val) noexcept;
integer::integer(short val) noexcept;
integer::integer(unsigned short val) noexcept;
integer::integer(int val) noexcept;
integer::integer(unsigned int val) noexcept;
integer::integer(long val) noexcept;
integer::integer(unsigned long val) noexcept;
integer::integer(long long val) noexcept;
integer::integer(unsigned long long val) noexcept;

integer::integer(float val);
integer::integer(double val);
integer::integer(long double val);

integer::integer(std::initializer_list<unspecified> list);

template<class CharT, class Traits, class Alloc>
    explicit integer::integer(const std::basic_string<CharT, Traits, Alloc>& str);

integer::integer(const bits& other);
integer::integer(bits&& other);

integer::integer(const integer& other);
integer::integer(integer&& other) noexcept;

The default constructor constructs an object whose value is 0.

The constructors that take integral arguments construct objects whose value is val.

The constructors that take floating-point arguments construct objects whose value is the value of val with any fractional part discarded.

The constructor that takes a string constructs an object whose value is the value represented by the string object. The string object shall have the form required for the string argument to the function std::strtol with a radix of base, and shall be interpreted as if by std::strtol(str.c_str(), 0, base), except that the resulting value can never be outside the range of representable values.

The constructor that takes an initializer_list constructs an object whose stored value is equal to the elements of the initializer_list treated as a series of unsigned 32-bit digits with the leftmost digit being most significant. For example, the initializer list { 0xFE, 0xF0, 0xAA, 0x31 } represents the value 0xFE * 323 + 0xF0 * 322 + 0xAA * 321 + 0x31 * 320.

The constructors that take an argument of type bits each construct an object whose stored value is the value in the bit pattern in other interpreted as a ones-complement representation of an integer value.

The copy and move constructors construct objects with the same value as other. The move constructor leaves other in an unspecified valid state.

bool integer::is_odd() const noexcept;

The member function returns true only if the stored value represents an odd number.

integer lcm(const integer& a, const integer& b);

The function returns an object whose value is the least common multiple of a and b.

integer mod(const integer& lhs, const integer& rhs);
integer& integer::mod(const integer& rhs);

The non-member function returns an object whose value is lhs mod rhs. The member function sets the stored value in *this to *this mod rhs and returns *this.

integer mulmod(const integer& lhs, const integer& rhs, const integer& m);
integer& integer::mulmod(const integer& rhs, const integer& m);

The non-member function returns an object whose value is (lhs * rhs) mod m. The member function sets the value of *this to (*this * rhs) mod m and returns *this.

integer& integer::negate() noexcept;

The member function sets the stored value of *this to the negation of its previous vaue and returns *this.

integer& integer::operator=(const integer& rhs);
integer& integer::operator=(integer&& rhs);

integer& integer::operator=(const bits& rhs);
integer& integer::operator=(bits&& rhs);

The first two operators store the value of rhs into *this. The last two operators store the value of rhs, interpreted as a ones-complement representation of an integer value, into *this. The operators all return *this.

integer operator+(const integer& lhs, const integer& rhs);
integer integer::operator+() const;

The first operator returns an object whose value is the sum of the values of lhs and rhs. The second operator returns a copy of *this.

integer& integer::operator+=(const integer& rhs);

The member operator sets the stored value of *this to the sum of the values of *this and rhs and returns a reference to *this.

integer& integer::operator++();
integer integer::operator++(int);

The member operators set the value stored in *this to *this + 1. The first operator returns *this. The second operator returns an object whose value is the value stored in *this prior to the increment.

integer operator-(const integer& lhs, const integer& rhs)
integer integer::operator-();

The first operator returns an object whose value is the difference between the values of lhs and rhs. The second operator returns an object whose value is the negation of the value of *this.

integer& integer::operator-=(const integer&);

The member operator sets the stored value of *this to the difference between the values of *this and rhs and returns *this.

integer& integer::operator--();
integer integer::operator--(int);

The member operators set the value stored in *this to *this - 1. The first operator returns *this. The second operator returns an object whose value is the value stored in *this prior to the decrement.

integer operator*(const integer& lhs, const integer& rhs);

The operator returns an object whose value is the product of the values of lhs and rhs.

integer& integer::operator*=(const integer& rhs);

The member operator sets the stored value of *this to the product of the values of *this and rhs and returns a reference to *this.

integer operator/(const integer& lhs, const integer& rhs);

The operator returns an object whose value is the quotient of the value of lhs divided by the value of rhs, discarding any fractional part.

integer& integer::operator/=(const integer& rhs);

The member operator sets the stored value of *this to the quotient of the value of *this divided by the value of rhs, discarding any fractional part, and returns *this.

integer operator%(const integer&, const integer&);

The operator returns an object whose value is the remainder of the value of lhs divided by the value of rhs. The remainder is the value such that (lhs / rhs) * rhs + lhs % rhs is equal to lhs.

integer& integer::operator%=(const integer&);

The member operator sets the stored value of *this to the remainder of *this divided by the value of rhs and returns *this.

integer operator>>(const integer& val, size_t rhs);

The operator returns an object whose value is val / 2rhs.

integer& integer::operator>>=(size_t rhs);

The operator sets the value of *this to *this / 2rhs. The operator returns *this.

template <class Elem, class Traits>
    std::basic_istream<Elem, Traits>& operator>>(std::basic_istream<Elem, Traits>& strm, integer& val);

The operator has the effect of { std::string temp; strm >> temp; val = integer(temp); }. It returns strm.

integer operator<<(const integer& val, size_t rhs);

The operator returns an object whose value is val * 2rhs.

integer& integer::operator<<=(size_t rhs);

The operator sets the value of *this to *this * 2rhs. The operator returns *this.

template <class Elem, class Traits>
    std::basic_ostream<Elem, Traits>& operator<<(std::basic_ostream<Elem, Traits>& strm, const integer& val);

The operator has the effect of strm << to_string(val). It returns strm.

explicit integer::operator bool() const noexcept;

The operator returns false only if *this is equal to 0.

explicit integer::operator long double() const noexcept;

The operator returns a value equal to the stored value of *this. If the stored value is outside the range that can be represented by an object of type long double the returned value is positive or negative infinity, as appropriate.

explicit integer::operator long long() const;

The operator returns a value equal to the stored value of *this. If the stored value cannot be represented as a long long it throws an exception of type range_error.

explicit integer::operator unsigned long long() const;

The operator returns a value equal to the stored value of *this. If the stored value cannot be represented as an unsigned long long it throws an exception of type range_error.

bool operator==(const integer& lhs, const integer& rhs) noexcept;

The operator returns true only if the value stored in lhs is equal to the value stored in rhs.

bool operator!=(const integer& lhs, const integer& rhs) noexcept;

The operator returns !(lhs == rhs).

bool operator>(const integer& lhs, const integer& rhs) noexcept;

The operator returns rhs < lhs.

bool operator>=(const integer& lhs, const integer& rhs) noexcept;

The operator returns !(lhs < rhs).

bool operator<(const integer& lhs, const integer& rhs) noexcept;

The operator return true only if lhs.compare(rhs) returns -1.

bool operator<=(const integer& lhs, const integer& rhs) noexcept;

The operator returns !(rhs < lhs).

integer pow(const integer& val, const integer& exp);
integer& integer::pow(const integer& exp);

The non-member function returns an object whose value is valexp. The member function sets the value of *this to *thisexp and returns *this. Requires: 0 <= exp.

integer powmod(const integer& val, const integer& exp, const integer& m);
integer& integer::powmod(const integer& exp, const integer& m);

The non-member function returns an object whose value is valexp mod m. The member function sets the value of *this to *thisexp mod m and returns *this. Requires: 0 <= exp and m != 0.

void integer::reserve(size_t digits);

The member function ensures that capacity() >= digits.

void integer::shrink_to_fit();

The member function is a non-binding request to reduce capacity() to hold the current stored value without wasted space.

size_t integer::size() const noexcept;

The member function returns capacity().

integer sqr(const integer& val);
integer& integer::sqr();

The non-member function returns an object whose value is val * val. The member function sets the value of *this to *this * *this and returns *this.

integer sqrt(const integer& val);
integer& integer::sqrt();

The non-member function returns an object whose value is the square root of the value held by val, discarding any fractional part. Requires: 0 <= val. The member function sets the value of *this to the square root of the value held by *this, discarding any fractional part, and returns *this. Requires: 0 <= *this.

void swap(integer& lhs, integer& rhs) noexcept;
void integer::swap(integer& rhs) noexcept;

The non-member function swaps the stored values of lhs and rhs. The member function swaps the stored values of *this and rhs.

std::string to_string(const integer& val, int radix = 10) const;

The function returns a string representation of the value stored in val, using radix as the radix.

Class integer_data_proxy

class integer_data_proxy {

    // type names
    typedef unspecified data_type;
    typedef unspecified arithmetic_type;
    typedef unspecified uarithmetic_type;
    typedef unspecified iterator;
    typedef unspecified const_iterator;
    typedef unspecified reverse_iterator;
    typedef unspecified const_reverse_iterator;

    // constructors
    integer_data_proxy(const integer_data_proxy& rhs) = delete;
    integer_data_proxy(integer_data_proxy&& rhs);

    // assign
    integer_data_proxy& operator=(const integer_data_proxy& rhs) = delete;
    integer_data_proxy& operator=(integer_data_proxy&& rhs) = delete;

    // iterators
    iterator begin() noexcept;
    iterator end() noexcept;
    reverse_iterator rbegin() noexcept;
    reverse_iterator rend() noexcept;
    const_iterator cbegin() const noexcept;
    const_iterator cend() const noexcept;
    const_reverse_iterator crbegin() const noexcept;
    const_reverse_iterator crend() const noexcept;

    // element access
    data_type operator[](size_t pos) const;
    data_type& operator[](size_t pos);

    // capacity
    size_t size() const noexcept;
    size_t capacity() const noexcept;
    void reserve(size_t digits);
    void shrink_to_fit();
};

The class describes an object that can be used to examine and modify the internal representation of an object of type integer. This allows advanced users to portably implement algorithms that are not provided natively.

There can be only one integer_data_proxy object associated with a particular integer object at any given time; that object is obtained by calling the get_data_proxy member function on the integer object. The resulting object can be moved but not copied.

typedef unspecified integer_data_proxy::arithmetic_type;

The typedef defines a synonym for a signed arithmetic type that is large enough to hold the product of the largest values that the implementation will store in an object of type data_type.

iterator integer_data_proxy::begin();

The member function returns an iterator object such that the iterators [begin(), end()) point to the internal data elements of the integer object.

size_t integer_data_proxy::capacity() const;

The member function returns the number of decimal digits that the integer object can represent without reallocating its internal storage.

const_iterator integer_data_proxy::cbegin() const;

The member function returns an iterator object such that the iterator range [cbegin(), cend()) points to the internal data elements of the integer object.

const_iterator integer_data_proxy::cend() const;

The member function returns an iterator object such that the iterator range [cbegin(), cend()) points to the internal data elements of the integer object.

typedef unspecified integer_data_proxy::const_iterator;

The typedef defines a synonym for an iterator that can be used to access but not modify internal data elements of the integer object.

typedef unspecified integer_data_proxy::const_reverse_iterator;

The typedef defines a synonym for a reverse iterator that can be used to access but not modify internal data elements of the integer object.

const_reverse_iterator integer_data_proxy::crbegin() const;

The member function returns a reverse iterator object such that the iterator range [crbegin(), crend()) points to the internal data elements of the integer object in reverse order.

const_reverse_iterator integer_data_proxy::crend() const;

The member function returns a reverse iterator object such that the iterator range [crbegin(), crend()) points to the internal data elements of the integer object in reverse order.

typedef unspecified integer_data_proxy::data_type;

The typedef defines a synonym for the type of the integer object's internal data elements.

iterator integer_data_proxy::end();

The member function returns an iterator object such that the iterator range [begin(), end()) points to the internal data elements of the integer object.

integer_data_proxy::integer_data_proxy(const integer_data_proxy&) = delete; 
integer_data_proxy::integer_data_proxy(integer_data_proxy&& rhs);

The copy constructor is deleted. The move constructor copies the contents of rhs and leaves rhs in an unspecified valid state.

typedef unspecified integer_data_proxy::iterator;

The typedef defines a synonym for an iterator that can be used to access internal data elements of the integer object.

integer& integer_data_proxy::operator=(const integer_data_proxy&) = delete;
integer& integer_data_proxy::operator=(integer_data_proxy&&) = delete;

The copy assignment and move assignment operators are deleted.

data_type integer_data_proxy::operator[](size_t pos) const;
data_type& integer_data_proxy::operator[](size_t pos);

The first member function returns the value of the internal data element at index pos. The second member function returns a reference to the internal data element at index pos.

reverse_iterator integer_data_proxy::rbegin();

The member function returns a reverse iterator object such that the iterator range [crbegin(), crend()) points to the internal data elements of the integer object in reverse order.

reverse_iterator integer_data_proxy::rend();

The member function returns a reverse iterator object such that the iterator range [crbegin(), crend()) points to the internal data elements of the integer object in reverse order.

void integer_data_proxy::reserve(size_t digits);

The member function ensures that capacity() >= digits.

typedef unspecified integer_data_proxy::reverse_iterator;

The typedef defines a synonym for a reverse iterator that can be used to access internal data elements of the integer object.

void integer_data_proxy::shrink_to_fit();

The member function is a non-binding request to reduce capacity() to hold the integer object's current stored value without wasted space.

size_t integer_data_proxy::size() const;

The member function returns capacity().

typedef unspecified integer_data:proxy::uarithmetic_type;

The typedef defines a synonym for an unsigned arithmetic type that is large enough to hold the product of the largest values that the implementation will store in an object of type data_type.

Class bits

class bits {
public:

    class reference;

    // constructors
    bits() noexcept;

    bits(char val) noexcept;
    bits(signed char val) noexcept;
    bits(unsigned char val) noexcept;
    bits(short val) noexcept;
    bits(unsigned short val) noexcept;
    bits(int val) noexcept;
    bits(unsigned val) noexcept;
    bits(long val) noexcept;
    bits(unsigned long val) noexcept;
    bits(long long val) noexcept;
    bits(unsigned long long val) noexcept;

    bits(std::initializer_list<uint_least32_t> list);

    template <class CharT, class Traits, class Alloc>
    explicit bits(const basic_string<CharT, Traits, Alloc>& str,
        typename basic_string<CharT, Traits, Alloc>::size_t pos = 0,
        typename basic_string<CharT, Traits, Alloc>::size_t count = std::basic_string<CharT>::npos,
        CharT zero = CharT('0'),
        CharT one = CharT('1'));
    template <class CharT>
    explicit bits(const CharT *ptr,
        typename basic_string<CharT>::size_t count = std::basic_string<CharT>::npos,
        CharT zero = CharT('0'),
        CharT one = CharT('1'));

    explicit bits(const integer& val);
    explicit bits(integer&& val);

    bits(const bits& other);
    bits(bits&& other) noexcept;

    // assign and swap
    bits& operator=(const bits& rhs);
    bits& operator=(bits&& rhs);
    bits& operator=(const integer& rhs);
    bits& operator=(integer&& rhs);
    void swap(bits& rhs) noexcept;

    // conversions
    unsigned long to_ulong() const;
    unsigned long long to_ullong() const;
    template <class CharT = char, class Traits = std::char_traits<CharT>, class Alloc = std::allocator<CharT> >
        std::basic_string<CharT, Traits, Alloc> to_string(CharT zero = CharT('0'), CharT one = CharT('1')) const;

    // logical operations
    bits& operator&=(const bits& rhs);
    bits& operator|=(const bits& rhs);
    bits& operator^=(const bits& rhs);
    bits operator~() const;
    
    bits& operator<<=(size_t rhs);
    bits& operator>>=(size_t rhs);
    bits& operator<<(size_t rhs) const;
    bits& operator>>(size_t rhs) const;

    // element access and modification
    bits& set() noexcept;
    bits& set(size_t pos, bool val = true);
    bits& reset() noexcept;
    bits& reset(size_t pos);
    bits& flip() noexcept;
    bits& flip(size_t pos);
    bool operator[](size_t pos) const;
    reference operator[](size_t pos);
    bool test(size_t pos) const noexcept;
    bool all() const noexcept;
    bool any() const noexcept;
    bool none() const noexcept;
    size_t count() const noexcept;
    size_t count_not_set() const noexcept;

    // comparison
    bool operator==(const bits& rhs) const noexcept;
    bool operator!=(const bits& rhs) const noexcept;

    // capacity
    size_t size() const noexcept;
    size_t capacity() const noexcept;
    void reserve(size_t bit_count);
    void shrink_to_fit();

};

The class describes an object that represents an unbounded set of bits.

bool bits::all() const noexcept

The member function returns true only if all the bits in *this are set.

bool bits::any() const noexcept

The member function returns true if at least one of the bits in *this is set.

bits::bits() noexcept;

bits::bits(char val) noexcept;
bits::bits(signed char val) noexcept;
bits::bits(unsigned char val) noexcept;
bits::bits(short val) noexcept;
bits::bits(unsigned short val) noexcept;
bits::bits(int val) noexcept;
bits::bits(unsigned int val) noexcept;
bits::bits(long val) noexcept;
bits::bits(unsigned long val) noexcept;
bits::bits(long long val) noexcept;
bits::bits(unsigned long long val) noexcept;

bits::bits(std::initializer_list<uint_least32_t> list);

template <class CharT, class Traits, class Alloc>
explicit bits::bits(const basic_string<CharT, Traits, Alloc>& str,
    typename basic_string<CharT, Traits, Alloc>::size_t pos = 0,
    typename basic_string<CharT, Traits, Alloc>::size_t count = std::basic_string<CharT>::npos,
    CharT zero = CharT('0'),
    CharT one = CharT('1'));
template <class CharT>
explicit bits::bits(const CharT *ptr,
    typename basic_string<CharT>::size_t count = std::basic_string<CharT>::npos,
    CharT zero = CharT('0'),
    CharT one = CharT('1'));

explicit bits::bits(const integer& val);
explicit bits::bits(integer&& val);

bits::bits(const bits& other);
bits::bits(bits&& other) noexcept;

The default constructor constructs an object whose value is 0.

The constructors that take integral arguments construct objects whose value is the ones-complement representation of val.

The constructor that takes an initializer_list constructs an object whose stored value is equal to the elements of the initializer_list treated as a series of unsigned 32-bit digits with the leftmost digit being most significant. For example, the initializer list { 0xFE, 0xF0, 0xAA, 0x31 } represents the value 0xFE * 323 + 0xF0 * 322 + 0xAA * 321 + 0x31 * 320.

The constructors that take string and const char* objects construct an object whose value is the value represented by their argument, treating zero as 0 and one as 1.

The constructors that take an argument of type integer construct objects whose value is the ones-complement representation of val.

The copy and move constructors construct objects with the same value as other. The move constructor leaves other in an unspecified valid state.

size_t bits::capacity() const noexcept;

The member function returns the number of bits that the object can represent without reallocating its internal storage.

size_t bits::count() const noexcept;

The member function returns the number of bits in *this that are set, or static_cast<size_t>(-1) if the number of bits that are set is too large to fit in an object of type size_t.

size_t bits::count_not_set() const noexcept;

The member function returns the number of bits in *this that are not set, or static_cast<size_t>(-1) if the number of bits that are not set is too large to fit in an object of type size_t.

void bits::flip() const noexcept;
void bits::flip(size_t pos);

The first member function toggles all the bits in the stored value. The second member function toggles the bit at position pos in the stored value.

bool bits::none() const noexcept;

The member function returns true only if none of the bits in *this is set.

bits& bits::operator=(const bits& rhs);
bits& bits::operator=(bits&& rhs);

bits& bits::operator=(const integer& rhs);
bits& bits::operator=(integer&& rhs);

The first two operators store the value of rhs into *this. The last two operators store the ones-complement representation of rhs into *this. All of the operators return *this.

bool bits::operator==(const bits& rhs) const noexcept;

The member operator returns true only if the stored value in *this is the same as the stored value in rhs.

bool bits::operator!=(const bits& rhs) const noexcept;

The member operator returns !(*this == rhs).

bits operator&(const bits& lhs, const bits& rhs);

The operator returns an object whose value is the bitwise AND of the values of lhs and rhs.

bits& bits::operator&=(const bits& rhs);

The member operator sets the value of *this to the bitwise AND of the values of *this and rhs and returns a reference to *this.

bits operator|(const bits& lhs, const bits& rhs);

The operator returns an object whose value is the bitwise inclusive OR of the values of lhs and rhs.

bits& bits::operator|=(const bits& rhs);

The member operator sets the value of *this to the bitwise inclusive OR of the values of *this and rhs and returns *this.

bits operator^(const bits& lhs, const bits& rhs);

The operator returns an object whose value is the bitwise exclusive OR of the values of lhs and rhs.

bits& bits::operator^=(const bits& rhs);

The member operator sets the value of *this to the bitwise exclusive OR of the values of *this and rhs and returns *this.

bits bits::operator~() const;

The member function returns an object that holds the complement of the set of bits held by *this.

bits operator>>(const bits& lhs, size_t rhs);

The operator returns an object whose stored value is the value of the bits in lhs shifted right rhs positions.

bits& bits::operator>>=(size_t rhs);

The operator sets the stored value in *this to the value of the bits in *this shifted right rhs positions. It returns *this.

template <class CharT, class Traits>
    std::basic_istream<CharT, Traits>& operator>>(
        std::basic_istream<CharT, Traits>& str, bits& val);

The operator has the effect of { std::string temp; strm >> temp; val = temp; }. It returns strm.

bits operator<<(const bits& lhs, size_t rhs);

The operator returns an object whose stored value is the value of the bits in lhs shifted left rhs positions.

bits& bits::operator<<=(size_t rhs);

The operator sets the stored value in *this to the value of the bits in *this shifted left rhs positions. It returns *this.

template <class CharT, class Traits>
    std::basic_ostream<CharT, Traits>& operator<<(
        std::basic_ostream<CharT, Traits>& str, const bits& val);

The operator has the effect of strm << val.to_string() and returns strm.

bool bits::operator[](size_t pos) const;
reference bits::operator[](size_t pos);

The first member function returns the value of the bit at position pos. The second member function returns an object of type bits::reference that refers to the bit at position pos.

void bits::reserve(size_t bit_count);

The member function ensures that capacity() >= bit_count.

bits& bits::reset() noexcept;
bits& bits::reset(size_t pos);

The first member function clears all the bits of *this. The second member function clears the bit as position pos. Both member functions return *this.

void bits::set() noexcept;
void bits::set(size_t pos, bool val = true);

The first member function sets all the bits of *this. The second member function sets the bit at position pos in the stored value to val. Both member functions return *this.

void bits::shrink_to_fit();

The member function is a non-binding request to reduce capacity() to hold the current stored value without wasted space.

size_t bits::size() const noexcept;

The member function returns capacity().

void swap(bits& lhs, bits& rhs) noexcept;
void bits::swap(bits& rhs) noexcept;

The non-member function swaps the stored values of lhs and rhs. The member function swaps the stored values of *this and rhs.

bool bits::test(size_t pos) const noexcept;

The member function returns true only if the bit at position pos in the stored value is non-zero.

template <class CharT = char, class Traits = std::char_traits<CharT>, class Alloc = std::allocator<CharT> >
    std::basic_string<CharT, Traits, Alloc> bits::to_string(
        CharT zero = CharT('0'), CharT one = CharT('1'));

The member function returns a string representation of the value stored in *this, using zero to represent 0 and one to represent 1.

unsigned long long bits::to_ullong() const;

The member function returns a value equal to the stored value of *this. It throws an exception of type range_error if the value cannot be represented as an unsigned long long.

unsigned long bits::to_ulong() const;

The member function returns a value equal to the stored value of *this. It throws an exception of type range_error if the value cannot be represented as a long long.

Class bits::reference

class bits {
    class reference {
    public:
        reference& operator=(bool val) noexcept;
        reference& operator=(const reference& rhs) noexcept;
        bool operator~() const noexcept;
        operator bool() const noexcept;
        reference& flip() noexcept;
    };
};

The nested class bits::reference describes an object that can be used to manage a particular bit in an object of type bits.

reference& bits::reference::flip() noexcept;

The member function toggles the bit that the object manages.

reference& bits::reference::operator=(bool rhs) noexcept;
reference& bits::reference::operator=(const reference& rhs) noexcept;

The first member operator sets the bit that the object manages to the value of rhs. The second member operator sets the bit that the object manages to the value managed by rhs.

bool bits::reference::operator~() const noexcept;

The member operator returns true if the bit managed by the object is set, otherwise false.

bits::reference::operator bool() const noexcept;

The member operator returns true if the bit that the object manages is set.