Arbitrary precision integer type. More...

#include <big_int.hpp>

Collaboration diagram for jau::mp::BigInt:

Public Types
enum	sign_t { negative = 0 , positive = 1 }
	Sign symbol definitions for positive and negative numbers. More...

Public Member Functions
	BigInt () noexcept=default

	BigInt (BigInt &&other) noexcept

	BigInt (const BigInt &o) noexcept=default

	BigInt (const std::string &str)
	Construct a big_int_t from a string encoded as hexadecimal or decimal. More...

	BigInt (const uint8_t buf[], size_t byte_len, const lb_endian_t byte_order)
	Create a big_int_t from an integer in a byte array with given byte_len, considering the given byte_order. More...

	BigInt (std::vector< mp_word_t > &&other_reg) noexcept

	BigInt (uint64_t n)
	Create big_int_t from an unsigned 64 bit integer. More...

	~BigInt () noexcept=default

BigInt	abs () const noexcept
	Returns absolute (positive) value of this instance. More...

size_t	binary_encode (uint8_t output[], size_t byte_len, const lb_endian_t byte_order) const noexcept
	Stores this number to the value in buf with given byte_len, considering the given byte_order. More...

size_t	bits () const noexcept
	Returns bit length of this integer. More...

uint8_t	byte_at (size_t n) const noexcept

size_t	bytes () const
	Returns byte length of this integer. More...

void	clear ()
	Zeroize the big_int. More...

int	compare (const BigInt &b) const noexcept
	Compares this instance against other considering both sign() value Returns. More...

void	conditionally_set_bit (size_t n, bool set_it) noexcept
	Conditionally set bit at specified position. More...

const mp_word_t *	data () const
	Return a const pointer to the register. More...

BigInt &	flip_sign () noexcept
	Flip the sign of this big_int. More...

bool	get_bit (size_t n) const noexcept
	Return bit value at specified position. More...

bool	is_even () const noexcept
	Test if the integer has an even value. More...

bool	is_negative () const noexcept
	Tests if the sign of the integer is negative. More...

bool	is_nonzero () const noexcept
	Test if the integer is not zero. More...

bool	is_odd () const noexcept
	Test if the integer has an odd value. More...

bool	is_positive () const noexcept
	Tests if the sign of the integer is positive. More...

bool	is_zero () const noexcept
	Test if the integer is zero. More...

BigInt &	mod_add (const BigInt &y, const BigInt &mod, std::vector< mp_word_t > &ws)
	Set this to (this + y) % mod This function assumes *this is >= 0 && < mod. More...

BigInt &	mod_mul (uint8_t y, const BigInt &mod, std::vector< mp_word_t > &ws)
	Set this to (this * y) % mod This function assumes *this is >= 0 && < mod y should be small, less than 16. More...

BigInt	mod_pow (BigInt e, BigInt m)
	Returns (this)^e % m, or pow(this, e) % m. More...

BigInt &	mod_sub (const BigInt &y, const BigInt &mod, std::vector< mp_word_t > &ws)
	Set this to (this - y) % mod This function assumes *this is >= 0 && < mod. More...

bool	operator! () const noexcept
	! operator, returns `true` iff this is zero, otherwise false. More...

bool	operator!= (const BigInt &b) const noexcept

BigInt	operator% (const BigInt &mod)

BigInt &	operator%= (const BigInt &mod)
	Modulo operator. More...

BigInt	operator* (const BigInt &y) noexcept

BigInt &	operator*= (const BigInt &y) noexcept

BigInt	operator+ (const BigInt &y) const noexcept

BigInt &	operator++ () noexcept

BigInt	operator++ (int) noexcept

BigInt &	operator+= (const BigInt &y) noexcept

BigInt	operator- () const noexcept
	Unary negation operator, returns new negative instance of this. More...

BigInt	operator- (const BigInt &y) const noexcept

BigInt &	operator-- () noexcept

BigInt	operator-- (int) noexcept

BigInt &	operator-= (const BigInt &y) noexcept

BigInt	operator/ (const BigInt &y) const

BigInt &	operator/= (const BigInt &y)

bool	operator< (const BigInt &b) const noexcept

BigInt	operator<< (size_t shift) const

BigInt &	operator<<= (size_t shift) noexcept

bool	operator<= (const BigInt &b) const noexcept

BigInt &	operator= (BigInt &&other) noexcept

BigInt &	operator= (const BigInt &r)=default

bool	operator== (const BigInt &b) const noexcept

bool	operator> (const BigInt &b) const noexcept

bool	operator>= (const BigInt &b) const noexcept

BigInt	operator>> (size_t shift) const

BigInt &	operator>>= (size_t shift) noexcept

BigInt	pow (BigInt e)
	Returns (this)^e, or pow(this, e) More...

BigInt &	rev_sub (const BigInt &y, std::vector< mp_word_t > &ws)
	Set this to y - this. More...

sign_t	reverse_sign () const noexcept

void	set_bit (size_t n) noexcept
	Set bit at specified position. More...

BigInt &	set_sign (sign_t sign) noexcept
	Set sign of the integer. More...

size_t	sig_words () const noexcept
	Return how many words we need to hold this value. More...

sign_t	sign () const noexcept
	Return the sign of the integer. More...

size_t	size () const noexcept
	Give size of internal register. More...

BigInt &	square (std::vector< mp_word_t > &ws)
	Square value of *this. More...

void	swap (BigInt &other) noexcept
	Swap this value with another. More...

std::string	to_dec_string (bool add_details=false) const noexcept

std::string	to_hex_string (bool add_details=false) const noexcept

mp_word_t	word_at (size_t n) const noexcept
	Return the mp_word_t at a specified position of the internal register. More...

Static Public Member Functions
static BigInt	from_s32 (int32_t n)
	Create big_int from a signed 32 bit integer. More...

static BigInt	from_u64 (uint64_t n)
	Create big_int from an unsigned 64 bit integer. More...

static BigInt	from_word (mp_word_t n)
	Create big_int from a mp_word_t (limb) More...

static BigInt	one ()
	Create a 1-value big_int. More...

static BigInt	power_of_2 (size_t n)
	Create a power of two. More...

static BigInt	with_capacity (size_t n)
	Create big_int of specified size, all zeros. More...

static BigInt	zero ()
	Create a 0-value big_int. More...

Detailed Description

Arbitrary precision integer type.

Local storage format

Internally the big integer is stored in an array of mp_word_t ordered little-endian alike, with the least significant word at the array-bottom and most significant word at the array-top.

The mp_word_t itself is stored in jau::endian::native!

Definition at line 43 of file big_int.hpp.

Member Enumeration Documentation

◆ sign_t

enum jau::mp::BigInt::sign_t

Sign symbol definitions for positive and negative numbers.

Enumerator
negative
positive

Definition at line 48 of file big_int.hpp.

Constructor & Destructor Documentation

◆ BigInt() [1/7]

jau::mp::BigInt::BigInt ( )

defaultnoexcept

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◆ BigInt() [2/7]

jau::mp::BigInt::BigInt ( const BigInt & o )

defaultnoexcept

◆ ~BigInt()

jau::mp::BigInt::~BigInt ( )

defaultnoexcept

◆ BigInt() [3/7]

jau::mp::BigInt::BigInt ( uint64_t n )

inline

Create big_int_t from an unsigned 64 bit integer.

Parameters

n	initial value of this big_int_t

Definition at line 129 of file big_int.hpp.

◆ BigInt() [4/7]

jau::mp::BigInt::BigInt ( const std::string & str )

inline

Construct a big_int_t from a string encoded as hexadecimal or decimal.

Both number bases may lead a -, denoting a negative number.

Hexadecimal is detected by a leading 0x.

Definition at line 145 of file big_int.hpp.

◆ BigInt() [5/7]

jau::mp::BigInt::BigInt	(	const uint8_t	buf[],
		size_t	byte_len,
		const lb_endian_t	byte_order
	)

inline

Create a big_int_t from an integer in a byte array with given byte_len, considering the given byte_order.

The value is stored in the local storage format, see bigint_storage_format

Parameters

buf	the byte array holding the value
byte_len	size of buf in bytes
littleEndian

Definition at line 178 of file big_int.hpp.

◆ BigInt() [6/7]

jau::mp::BigInt::BigInt ( std::vector< mp_word_t > && other_reg )

inlinenoexcept

Definition at line 182 of file big_int.hpp.

◆ BigInt() [7/7]

jau::mp::BigInt::BigInt ( BigInt && other )

inlinenoexcept

Definition at line 186 of file big_int.hpp.

Member Function Documentation

◆ zero()

static BigInt jau::mp::BigInt::zero ( )

inlinestatic

Create a 0-value big_int.

Definition at line 60 of file big_int.hpp.

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◆ one()

static BigInt jau::mp::BigInt::one ( )

inlinestatic

Create a 1-value big_int.

Definition at line 65 of file big_int.hpp.

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◆ from_u64()

static BigInt jau::mp::BigInt::from_u64 ( uint64_t n )

inlinestatic

Create big_int from an unsigned 64 bit integer.

Parameters

n	initial value of this big_int

Definition at line 71 of file big_int.hpp.

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◆ from_word()

static BigInt jau::mp::BigInt::from_word ( mp_word_t n )

inlinestatic

Create big_int from a mp_word_t (limb)

Parameters

n	initial value of this big_int

Definition at line 86 of file big_int.hpp.

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◆ from_s32()

static BigInt jau::mp::BigInt::from_s32 ( int32_t n )

inlinestatic

Create big_int from a signed 32 bit integer.

Parameters

n	initial value of this big_int

Definition at line 96 of file big_int.hpp.

◆ with_capacity()

static BigInt jau::mp::BigInt::with_capacity ( size_t n )

inlinestatic

Create big_int of specified size, all zeros.

Parameters

n	size of the internal register in words

Definition at line 108 of file big_int.hpp.

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◆ power_of_2()

static BigInt jau::mp::BigInt::power_of_2 ( size_t n )

inlinestatic

Create a power of two.

Parameters

n	the power of two to create

Returns: big_int_t representing 2^n

Definition at line 119 of file big_int.hpp.

◆ operator=() [1/2]

BigInt & jau::mp::BigInt::operator= ( const BigInt & r )

default

◆ operator=() [2/2]

BigInt & jau::mp::BigInt::operator= ( BigInt && other )

inlinenoexcept

Definition at line 192 of file big_int.hpp.

◆ swap()

void jau::mp::BigInt::swap ( BigInt & other )

inlinenoexcept

Swap this value with another.

Parameters

other big_int to swap values with

Definition at line 203 of file big_int.hpp.

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◆ operator-() [1/2]

BigInt jau::mp::BigInt::operator- ( ) const

inlinenoexcept

Unary negation operator, returns new negative instance of this.

Definition at line 216 of file big_int.hpp.

◆ byte_at()

uint8_t jau::mp::BigInt::byte_at ( size_t n ) const

inlinenoexcept

Parameters

n	the offset to get a byte from

Returns: byte at offset n

Definition at line 222 of file big_int.hpp.

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◆ word_at()

mp_word_t jau::mp::BigInt::word_at ( size_t n ) const

inlinenoexcept

Return the mp_word_t at a specified position of the internal register.

Parameters

n	position in the register

Returns: value at position n

Definition at line 232 of file big_int.hpp.

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◆ data()

const mp_word_t * jau::mp::BigInt::data ( ) const

inline

Return a const pointer to the register.

Returns: a pointer to the start of the internal register

Definition at line 240 of file big_int.hpp.

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◆ is_negative()

bool jau::mp::BigInt::is_negative ( ) const

inlinenoexcept

Tests if the sign of the integer is negative.

Returns: true, iff the integer has a negative sign

Definition at line 246 of file big_int.hpp.

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◆ is_positive()

bool jau::mp::BigInt::is_positive ( ) const

inlinenoexcept

Tests if the sign of the integer is positive.

Returns: true, iff the integer has a positive sign

Definition at line 252 of file big_int.hpp.

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◆ sign()

sign_t jau::mp::BigInt::sign ( ) const

inlinenoexcept

Return the sign of the integer.

Returns: the sign of the integer

Definition at line 258 of file big_int.hpp.

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◆ reverse_sign()

sign_t jau::mp::BigInt::reverse_sign ( ) const

inlinenoexcept

Returns: the opposite sign of the represented integer value

Definition at line 263 of file big_int.hpp.

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◆ flip_sign()

BigInt & jau::mp::BigInt::flip_sign ( )

inlinenoexcept

Flip the sign of this big_int.

Definition at line 273 of file big_int.hpp.

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◆ set_sign()

BigInt & jau::mp::BigInt::set_sign ( sign_t sign )

inlinenoexcept

Set sign of the integer.

Parameters

sign	new Sign to set

Definition at line 281 of file big_int.hpp.

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◆ abs()

BigInt jau::mp::BigInt::abs ( ) const

inlinenoexcept

Returns absolute (positive) value of this instance.

Definition at line 290 of file big_int.hpp.

◆ size()

size_t jau::mp::BigInt::size ( ) const

inlinenoexcept

Give size of internal register.

Returns: size of internal register in words

Definition at line 298 of file big_int.hpp.

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◆ sig_words()

size_t jau::mp::BigInt::sig_words ( ) const

inlinenoexcept

Return how many words we need to hold this value.

Returns: significant words of the represented integer value

Definition at line 304 of file big_int.hpp.

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◆ bytes()

size_t jau::mp::BigInt::bytes ( ) const

inline

Returns byte length of this integer.

Definition at line 307 of file big_int.hpp.

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◆ bits()

size_t jau::mp::BigInt::bits ( ) const

inlinenoexcept

Returns bit length of this integer.

Definition at line 310 of file big_int.hpp.

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◆ clear()

void jau::mp::BigInt::clear ( )

inline

Zeroize the big_int.

The size of the underlying register is not modified.

Definition at line 324 of file big_int.hpp.

◆ compare()

int jau::mp::BigInt::compare ( const BigInt & b ) const

inlinenoexcept

Compares this instance against other considering both sign() value Returns.

-1 if this < other
0 if this == other
1 if this > other

Definition at line 333 of file big_int.hpp.

◆ is_even()

bool jau::mp::BigInt::is_even ( ) const

inlinenoexcept

Test if the integer has an even value.

Returns: true if the integer is even, false otherwise

Definition at line 341 of file big_int.hpp.

◆ is_odd()

bool jau::mp::BigInt::is_odd ( ) const

inlinenoexcept

Test if the integer has an odd value.

Returns: true if the integer is odd, false otherwise

Definition at line 347 of file big_int.hpp.

◆ is_nonzero()

bool jau::mp::BigInt::is_nonzero ( ) const

inlinenoexcept

Test if the integer is not zero.

Returns: true if the integer is non-zero, false otherwise

Definition at line 353 of file big_int.hpp.

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◆ is_zero()

bool jau::mp::BigInt::is_zero ( ) const

inlinenoexcept

Test if the integer is zero.

Returns: true if the integer is zero, false otherwise

Definition at line 359 of file big_int.hpp.

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◆ get_bit()

bool jau::mp::BigInt::get_bit ( size_t n ) const

inlinenoexcept

Return bit value at specified position.

Parameters

n	the bit offset to test

Returns: true, if the bit at position n is set, false otherwise

Definition at line 368 of file big_int.hpp.

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◆ set_bit()

void jau::mp::BigInt::set_bit ( size_t n )

inlinenoexcept

Set bit at specified position.

Parameters

n	bit position to set

Definition at line 376 of file big_int.hpp.

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◆ conditionally_set_bit()

void jau::mp::BigInt::conditionally_set_bit	(	size_t	n,
		bool	set_it
	)

inlinenoexcept

Conditionally set bit at specified position.

Note if set_it is false, nothing happens, and if the bit is already set, it remains set.

Parameters

n	bit position to set
set_it	if the bit should be set

Definition at line 388 of file big_int.hpp.

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◆ operator!()

bool jau::mp::BigInt::operator! ( ) const

inlinenoexcept

! operator, returns true iff this is zero, otherwise false.

Definition at line 395 of file big_int.hpp.

◆ operator==()

bool jau::mp::BigInt::operator== ( const BigInt & b ) const

inlinenoexcept

Definition at line 397 of file big_int.hpp.

◆ operator!=()

bool jau::mp::BigInt::operator!= ( const BigInt & b ) const

inlinenoexcept

Definition at line 398 of file big_int.hpp.

◆ operator<=()

bool jau::mp::BigInt::operator<= ( const BigInt & b ) const

inlinenoexcept

Definition at line 399 of file big_int.hpp.

◆ operator>=()

bool jau::mp::BigInt::operator>= ( const BigInt & b ) const

inlinenoexcept

Definition at line 400 of file big_int.hpp.

◆ operator<()

bool jau::mp::BigInt::operator< ( const BigInt & b ) const

inlinenoexcept

Definition at line 401 of file big_int.hpp.

◆ operator>()

bool jau::mp::BigInt::operator> ( const BigInt & b ) const

inlinenoexcept

Definition at line 402 of file big_int.hpp.

◆ operator++() [1/2]

BigInt & jau::mp::BigInt::operator++ ( )

inlinenoexcept

Definition at line 410 of file big_int.hpp.

◆ operator--() [1/2]

BigInt & jau::mp::BigInt::operator-- ( )

inlinenoexcept

Definition at line 412 of file big_int.hpp.

◆ operator++() [2/2]

BigInt jau::mp::BigInt::operator++ ( int )

inlinenoexcept

Definition at line 414 of file big_int.hpp.

◆ operator--() [2/2]

BigInt jau::mp::BigInt::operator-- ( int )

inlinenoexcept

Definition at line 416 of file big_int.hpp.

◆ operator+=()

BigInt & jau::mp::BigInt::operator+= ( const BigInt & y )

inlinenoexcept

Definition at line 418 of file big_int.hpp.

◆ operator-=()

BigInt & jau::mp::BigInt::operator-= ( const BigInt & y )

inlinenoexcept

Definition at line 422 of file big_int.hpp.

◆ operator+()

BigInt jau::mp::BigInt::operator+ ( const BigInt & y ) const

inlinenoexcept

Definition at line 426 of file big_int.hpp.

◆ operator-() [2/2]

BigInt jau::mp::BigInt::operator- ( const BigInt & y ) const

inlinenoexcept

Definition at line 430 of file big_int.hpp.

◆ operator<<=()

BigInt & jau::mp::BigInt::operator<<= ( size_t shift )

inlinenoexcept

Definition at line 434 of file big_int.hpp.

◆ operator>>=()

BigInt & jau::mp::BigInt::operator>>= ( size_t shift )

inlinenoexcept

Definition at line 450 of file big_int.hpp.

◆ operator<<()

BigInt jau::mp::BigInt::operator<< ( size_t shift ) const

inline

Definition at line 462 of file big_int.hpp.

◆ operator>>()

BigInt jau::mp::BigInt::operator>> ( size_t shift ) const

inline

Definition at line 473 of file big_int.hpp.

◆ operator*=()

BigInt & jau::mp::BigInt::operator*= ( const BigInt & y )

inlinenoexcept

Definition at line 494 of file big_int.hpp.

◆ operator*()

BigInt jau::mp::BigInt::operator* ( const BigInt & y )

inlinenoexcept

Definition at line 499 of file big_int.hpp.

◆ operator/=()

BigInt & jau::mp::BigInt::operator/= ( const BigInt & y )

inline

Definition at line 518 of file big_int.hpp.

◆ operator/()

BigInt jau::mp::BigInt::operator/ ( const BigInt & y ) const

inline

Definition at line 527 of file big_int.hpp.

◆ operator%=()

BigInt & jau::mp::BigInt::operator%= ( const BigInt & mod )

inline

Modulo operator.

Parameters

y	the modulus to reduce this by

Definition at line 539 of file big_int.hpp.

◆ operator%()

BigInt jau::mp::BigInt::operator% ( const BigInt & mod )

inline

Definition at line 543 of file big_int.hpp.

◆ pow()

BigInt jau::mp::BigInt::pow ( BigInt e )

inline

Returns (*this)^e, or pow(*this, e)

Implementation is not optimized and naive, i.e. O(n)

Parameters

e	the exponent

Definition at line 568 of file big_int.hpp.

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◆ mod_pow()

BigInt jau::mp::BigInt::mod_pow	(	BigInt	e,
		BigInt	m
	)

inline

Returns (*this)^e % m, or pow(*this, e) % m.

Implementation is not optimized and naive, i.e. O(n)

Parameters

e	the exponent

Definition at line 602 of file big_int.hpp.

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◆ square()

BigInt & jau::mp::BigInt::square ( std::vector< mp_word_t > & ws )

Square value of *this.

Parameters

ws	a temp workspace

◆ rev_sub()

BigInt & jau::mp::BigInt::rev_sub	(	const BigInt &	y,
		std::vector< mp_word_t > &	ws
	)

Set *this to y - *this.

Parameters

y	the big_int_t to subtract from
ws	a temp workspace

◆ mod_add()

BigInt & jau::mp::BigInt::mod_add	(	const BigInt &	y,
		const BigInt &	mod,
		std::vector< mp_word_t > &	ws
	)

Set *this to (*this + y) % mod This function assumes *this is >= 0 && < mod.

Parameters

y	the big_int_t to add - assumed y >= 0 and y < mod
mod	the positive modulus
ws	a temp workspace

◆ mod_sub()

BigInt & jau::mp::BigInt::mod_sub	(	const BigInt &	y,
		const BigInt &	mod,
		std::vector< mp_word_t > &	ws
	)

Set *this to (*this - y) % mod This function assumes *this is >= 0 && < mod.

Parameters

y	the big_int_t to subtract - assumed y >= 0 and y < mod
mod	the positive modulus
ws	a temp workspace

◆ mod_mul()

BigInt & jau::mp::BigInt::mod_mul	(	uint8_t	y,
		const BigInt &	mod,
		std::vector< mp_word_t > &	ws
	)

Set *this to (*this * y) % mod This function assumes *this is >= 0 && < mod y should be small, less than 16.

Parameters

y	the small integer to multiply by
mod	the positive modulus
ws	a temp workspace

◆ to_dec_string()

std::string jau::mp::BigInt::to_dec_string ( bool add_details = false ) const

inlinenoexcept

Parameters

rng	a random number generator
min	the minimum value (must be non-negative)
max	the maximum value (must be non-negative and > min)

Returns: random integer in [min,max) static big_int_t random_integer(RandomNumberGenerator& rng, const big_int_t& min, const big_int_t& max); // TODO

Definition at line 681 of file big_int.hpp.

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◆ to_hex_string()

std::string jau::mp::BigInt::to_hex_string ( bool add_details = false ) const

inlinenoexcept

Definition at line 754 of file big_int.hpp.

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◆ binary_encode()

size_t jau::mp::BigInt::binary_encode	(	uint8_t	output[],
		size_t	byte_len,
		const lb_endian_t	byte_order
	)		const

inlinenoexcept

Stores this number to the value in buf with given byte_len, considering the given byte_order.

The value is read from the local storage in its format, see bigint_storage_format

If byte_len is less than the byt-esize of this integer, i.e. bytes(), then it will be truncated.

If byte_len is greater than the byte-size of this integer, i.e. bytes(), it will be zero-padded.

Returns: actual number of bytes copied, i.e. min(byte_len, bytes());

Definition at line 1119 of file big_int.hpp.

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The documentation for this class was generated from the following file:

jaulib/include/jau/mp/big_int.hpp

Public Types

Public Member Functions

Static Public Member Functions

Detailed Description

Local storage format

Member Enumeration Documentation

◆ sign_t

Constructor & Destructor Documentation

◆ BigInt() [1/7]

◆ BigInt() [2/7]

◆ ~BigInt()

◆ BigInt() [3/7]

◆ BigInt() [4/7]

◆ BigInt() [5/7]

◆ BigInt() [6/7]

◆ BigInt() [7/7]

Member Function Documentation

◆ zero()

◆ one()

◆ from_u64()

◆ from_word()

◆ from_s32()

◆ with_capacity()

◆ power_of_2()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ swap()

◆ operator-() [1/2]

◆ byte_at()

◆ word_at()

◆ data()

◆ is_negative()

◆ is_positive()

◆ sign()

◆ reverse_sign()

◆ flip_sign()

◆ set_sign()

◆ abs()

◆ size()

◆ sig_words()

◆ bytes()

◆ bits()

◆ clear()

◆ compare()

◆ is_even()

◆ is_odd()

◆ is_nonzero()

◆ is_zero()

◆ get_bit()

◆ set_bit()

◆ conditionally_set_bit()

◆ operator!()

◆ operator==()

◆ operator!=()

◆ operator<=()

◆ operator>=()

◆ operator<()

◆ operator>()

◆ operator++() [1/2]

◆ operator--() [1/2]

◆ operator++() [2/2]

◆ operator--() [2/2]

◆ operator+=()

◆ operator-=()

◆ operator+()

◆ operator-() [2/2]

◆ operator<<=()

◆ operator>>=()

◆ operator<<()

◆ operator>>()

◆ operator*=()

◆ operator*()

◆ operator/=()

◆ operator/()

◆ operator%=()

◆ operator%()

◆ pow()

◆ mod_pow()

◆ square()

◆ rev_sub()