1
0
mirror of https://github.com/vlang/v.git synced 2023-08-10 21:13:21 +03:00
v/vlib/math/big/integer.v

1093 lines
26 KiB
V
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

module big
import math.bits
import strings
import strconv
const digit_array = '0123456789abcdefghijklmnopqrstuvwxyz'.bytes()
// big.Integer
// -----------
// It has the following properties:
// 1. Every "digit" is an integer in the range [0, 2^32).
// 2. The signum can be one of three values: -1, 0, +1 for
// negative, zero, and positive values, respectively.
// 3. There should be no leading zeros in the digit array.
// 4. The digits are stored in little endian format, that is,
// the digits with a lower positional value (towards the right
// when represented as a string) have a lower index, and vice versa.
pub struct Integer {
digits []u32
pub:
signum int
is_const bool
}
[unsafe]
fn (mut x Integer) free() {
if x.is_const {
return
}
unsafe { x.digits.free() }
}
fn (x Integer) clone() Integer {
return Integer{
digits: x.digits.clone()
signum: x.signum
is_const: false
}
}
fn int_signum(value int) int {
if value == 0 {
return 0
}
return if value < 0 { -1 } else { 1 }
}
// integer_from_int creates a new `big.Integer` from the given int value.
pub fn integer_from_int(value int) Integer {
if value == 0 {
return zero_int
}
return Integer{
digits: [u32(iabs(value))]
signum: int_signum(value)
}
}
// integer_from_u32 creates a new `big.Integer` from the given u32 value.
pub fn integer_from_u32(value u32) Integer {
if value == 0 {
return zero_int
}
return Integer{
digits: [value]
signum: 1
}
}
// integer_from_i64 creates a new `big.Integer` from the given i64 value.
pub fn integer_from_i64(value i64) Integer {
if value == 0 {
return zero_int
}
signum_value := if value < 0 { -1 } else { 1 }
abs_value := u64(value * signum_value)
lower := u32(abs_value)
upper := u32(abs_value >> 32)
if upper == 0 {
return Integer{
digits: [lower]
signum: signum_value
}
} else {
return Integer{
digits: [lower, upper]
signum: signum_value
}
}
}
// integer_from_u64 creates a new `big.Integer` from the given u64 value.
pub fn integer_from_u64(value u64) Integer {
if value == 0 {
return zero_int
}
lower := u32(value & 0x00000000ffffffff)
upper := u32((value & 0xffffffff00000000) >> 32)
if upper == 0 {
return Integer{
digits: [lower]
signum: 1
}
} else {
return Integer{
digits: [lower, upper]
signum: 1
}
}
}
[params]
pub struct IntegerConfig {
signum int = 1
}
// integer_from_bytes creates a new `big.Integer` from the given byte array.
// By default, positive integers are assumed.
// If you want a negative integer, use in the following manner:
// `value := big.integer_from_bytes(bytes, signum: -1)`
[direct_array_access]
pub fn integer_from_bytes(input []u8, config IntegerConfig) Integer {
// Thank you to Miccah (@mcastorina) for this implementation and relevant unit tests.
if input.len == 0 {
return integer_from_int(0)
}
// pad input
mut padded_input := []u8{len: ((input.len + 3) & ~0x3) - input.len, cap: (input.len + 3) & ~0x3}
padded_input << input
mut digits := []u32{len: padded_input.len / 4}
// combine every 4 bytes into a u32 and insert into n.digits
for i := 0; i < padded_input.len; i += 4 {
x3 := u32(padded_input[i])
x2 := u32(padded_input[i + 1])
x1 := u32(padded_input[i + 2])
x0 := u32(padded_input[i + 3])
val := (x3 << 24) | (x2 << 16) | (x1 << 8) | x0
digits[(padded_input.len - i) / 4 - 1] = val
}
return Integer{
digits: digits
signum: config.signum
}
}
// integer_from_string creates a new `big.Integer` from the decimal digits specified in the given string.
// For other bases, use `big.integer_from_radix` instead.
pub fn integer_from_string(characters string) !Integer {
return integer_from_radix(characters, 10)
}
// integer_from_radix creates a new `big.Integer` from the given string and radix.
pub fn integer_from_radix(all_characters string, radix u32) !Integer {
if radix < 2 || radix > 36 {
return error('Radix must be between 2 and 36 (inclusive)')
}
characters := all_characters.to_lower()
validate_string(characters, radix)!
return match radix {
2 {
integer_from_special_string(characters, 1)
}
16 {
integer_from_special_string(characters, 4)
}
else {
integer_from_regular_string(characters, radix)
}
}
}
[direct_array_access]
fn validate_string(characters string, radix u32) ! {
sign_present := characters[0] == `+` || characters[0] == `-`
start_index := if sign_present { 1 } else { 0 }
for index := start_index; index < characters.len; index++ {
digit := characters[index]
value := big.digit_array.index(digit)
if value == -1 {
return error('Invalid character ${digit}')
}
if value >= radix {
return error('Invalid character ${digit} for base ${radix}')
}
}
}
[direct_array_access]
fn integer_from_special_string(characters string, chunk_size int) Integer {
sign_present := characters[0] == `+` || characters[0] == `-`
signum := if sign_present {
if characters[0] == `-` { -1 } else { 1 }
} else {
1
}
start_index := if sign_present { 1 } else { 0 }
mut big_digits := []u32{cap: ((characters.len * chunk_size) >> 5) + 1}
mut current := u32(0)
mut offset := 0
for index := characters.len - 1; index >= start_index; index-- {
digit := characters[index]
value := u32(big.digit_array.index(digit))
current |= value << offset
offset += chunk_size
if offset == 32 {
big_digits << current
current = u32(0)
offset = 0
}
}
// Store the accumulated value into the digit array
if current != 0 {
big_digits << current
}
shrink_tail_zeros(mut big_digits)
return Integer{
digits: big_digits
signum: if big_digits.len == 0 { 0 } else { signum }
}
}
[direct_array_access]
fn integer_from_regular_string(characters string, radix u32) Integer {
sign_present := characters[0] == `+` || characters[0] == `-`
signum := if sign_present {
if characters[0] == `-` { -1 } else { 1 }
} else {
1
}
start_index := if sign_present { 1 } else { 0 }
mut result := zero_int
radix_int := integer_from_u32(radix)
for index := start_index; index < characters.len; index++ {
digit := characters[index]
value := big.digit_array.index(digit)
result *= radix_int
result += integer_from_int(value)
}
return Integer{
digits: result.digits.clone()
signum: result.signum * signum
}
}
// abs returns the absolute value of the integer.
pub fn (integer Integer) abs() Integer {
return if integer.signum == 0 {
zero_int
} else {
Integer{
digits: integer.digits.clone()
signum: 1
}
}
}
// neg returns the result of negation of the integer.
pub fn (integer Integer) neg() Integer {
return if integer.signum == 0 {
zero_int
} else {
Integer{
digits: integer.digits.clone()
signum: -integer.signum
}
}
}
pub fn (integer Integer) + (addend Integer) Integer {
// Quick exits
if integer.signum == 0 {
return addend.clone()
}
if addend.signum == 0 {
return integer.clone()
}
// Non-zero cases
return if integer.signum == addend.signum {
integer.add(addend)
} else { // Unequal signs
integer.subtract(addend)
}
}
pub fn (integer Integer) - (subtrahend Integer) Integer {
// Quick exits
if integer.signum == 0 {
return subtrahend.neg()
}
if subtrahend.signum == 0 {
return integer.clone()
}
// Non-zero cases
return if integer.signum == subtrahend.signum {
integer.subtract(subtrahend)
} else {
integer.add(subtrahend)
}
}
fn (integer Integer) add(addend Integer) Integer {
a := integer.digits
b := addend.digits
mut storage := []u32{len: imax(a.len, b.len) + 1}
add_digit_array(a, b, mut storage)
return Integer{
signum: integer.signum
digits: storage
}
}
fn (integer Integer) subtract(subtrahend Integer) Integer {
cmp := integer.abs_cmp(subtrahend)
if cmp == 0 {
return zero_int
}
a, b := if cmp > 0 { integer, subtrahend } else { subtrahend, integer }
mut storage := []u32{len: a.digits.len}
subtract_digit_array(a.digits, b.digits, mut storage)
return Integer{
signum: cmp * a.signum
digits: storage
}
}
pub fn (integer Integer) * (multiplicand Integer) Integer {
// Quick exits
if integer.signum == 0 || multiplicand.signum == 0 {
return zero_int
}
if integer == one_int {
return multiplicand.clone()
}
if multiplicand == one_int {
return integer.clone()
}
// The final sign is the product of the signs
mut storage := []u32{len: integer.digits.len + multiplicand.digits.len}
multiply_digit_array(integer.digits, multiplicand.digits, mut storage)
return Integer{
signum: integer.signum * multiplicand.signum
digits: storage
}
}
// div_mod returns the quotient and remainder of the integer division.
pub fn (integer Integer) div_mod(divisor Integer) (Integer, Integer) {
// Quick exits
if divisor.signum == 0 {
panic('Cannot divide by zero')
}
if integer.signum == 0 {
return zero_int, zero_int
}
if divisor == one_int {
return integer.clone(), zero_int
}
if divisor.signum == -1 {
q, r := integer.div_mod(divisor.neg())
return q.neg(), r
}
if integer.signum == -1 {
q, r := integer.neg().div_mod(divisor)
if r.signum == 0 {
return q.neg(), zero_int
} else {
return q.neg() - one_int, divisor - r
}
}
// Division for positive integers
mut q := []u32{cap: integer.digits.len - divisor.digits.len + 1}
mut r := []u32{cap: integer.digits.len}
divide_digit_array(integer.digits, divisor.digits, mut q, mut r)
quotient := Integer{
signum: if q.len == 0 { 0 } else { 1 }
digits: q
}
remainder := Integer{
signum: if r.len == 0 { 0 } else { 1 }
digits: r
}
return quotient, remainder
}
pub fn (a Integer) / (b Integer) Integer {
q, _ := a.div_mod(b)
return q
}
pub fn (a Integer) % (b Integer) Integer {
_, r := a.div_mod(b)
return r
}
// mask_bits is the equivalent of `a % 2^n` (only when `a >= 0`), however doing a full division
// run for this would be a lot of work when we can simply "cut off" all bits to the left of
// the `n`th bit.
[direct_array_access]
fn (a Integer) mask_bits(n u32) Integer {
$if debug {
assert a.signum >= 0
}
if a.digits.len == 0 || n == 0 {
return zero_int
}
w := n / 32
b := n % 32
if w >= a.digits.len {
return a
}
return Integer{
digits: if b == 0 {
mut storage := []u32{len: int(w)}
for i := 0; i < storage.len; i++ {
storage[i] = a.digits[i]
}
storage
} else {
mut storage := []u32{len: int(w) + 1}
for i := 0; i < storage.len; i++ {
storage[i] = a.digits[i]
}
storage[w] &= ~(u32(-1) << b)
storage
}
signum: 1
}
}
// pow returns the integer `a` raised to the power of the u32 `exponent`.
pub fn (base Integer) pow(exponent u32) Integer {
if exponent == 0 {
return one_int
}
if exponent == 1 {
return base.clone()
}
mut n := exponent
mut x := base
mut y := one_int
for n > 1 {
if n & 1 == 1 {
y *= x
}
x *= x
n >>= 1
}
return x * y
}
// mod_pow returns the integer `a` raised to the power of the u32 `exponent` modulo the integer `modulus`.
pub fn (base Integer) mod_pow(exponent u32, modulus Integer) Integer {
if exponent == 0 {
return one_int
}
if exponent == 1 {
return base % modulus
}
mut n := exponent
mut x := base % modulus
mut y := one_int
for n > 1 {
if n & 1 == 1 {
y *= x % modulus
}
x *= x % modulus
n >>= 1
}
return x * y % modulus
}
// big_mod_pow returns the integer `base` raised to the power of the integer `exponent` modulo the integer `modulus`.
[direct_array_access]
pub fn (base Integer) big_mod_pow(exponent Integer, modulus Integer) !Integer {
if exponent.signum < 0 {
return error('math.big: Exponent needs to be non-negative.')
}
// this goes first as otherwise 1 could be returned incorrectly if base == 1
if modulus.bit_len() <= 1 {
return zero_int
}
// x^0 == 1 || 1^x == 1
if exponent.signum == 0 || base.bit_len() == 1 {
return one_int
}
// 0^x == 0 (x != 0 due to previous clause)
if base.signum == 0 {
return one_int
}
if exponent.bit_len() == 1 {
// x^1 without mod == x
if modulus.signum == 0 {
return base
}
// x^1 (mod m) === x % m
return base % modulus
}
// the amount of precomputation in windowed exponentiation (done in the montgomery and binary
// windowed exponentiation algorithms) is far too costly for small sized exponents, so
// we redirect the call to mod_pow
return if exponent.digits.len > 1 {
if modulus.is_odd() {
// modulus is odd, therefore we use the normal
// montgomery modular exponentiation algorithm
base.mont_odd(exponent, modulus)
} else if modulus.is_power_of_2() {
base.exp_binary(exponent, modulus)
} else {
base.mont_even(exponent, modulus)
}
} else {
base.mod_pow(exponent.digits[0], modulus)
}
}
// inc returns the integer `a` incremented by 1.
pub fn (mut a Integer) inc() {
a = a + one_int
}
// dec returns the integer `a` decremented by 1.
pub fn (mut a Integer) dec() {
a = a - one_int
}
pub fn (a Integer) == (b Integer) bool {
return a.signum == b.signum && a.digits.len == b.digits.len && a.digits == b.digits
}
// abs_cmp returns the result of comparing the magnitudes of the integers `a` and `b`.
// It returns a negative int if `|a| < |b|`, 0 if `|a| == |b|`, and a positive int if `|a| > |b|`.
pub fn (a Integer) abs_cmp(b Integer) int {
return compare_digit_array(a.digits, b.digits)
}
pub fn (a Integer) < (b Integer) bool {
// Quick exits based on signum value:
if a.signum < b.signum {
return true
}
if a.signum > b.signum {
return false
}
// They have equal sign
signum := a.signum
if signum == 0 { // Are they both zero?
return false
}
// If they are negative, the one with the larger absolute value is smaller
cmp := a.abs_cmp(b)
return if signum < 0 { cmp > 0 } else { cmp < 0 }
}
fn check_sign(a Integer) {
if a.signum < 0 {
panic('Bitwise operations are only supported for nonnegative integers')
}
}
// get_bit checks whether the bit at the given index is set.
[direct_array_access]
pub fn (a Integer) get_bit(i u32) bool {
check_sign(a)
target_index := i / 32
offset := i % 32
if target_index >= a.digits.len {
return false
}
return (a.digits[target_index] >> offset) & 1 != 0
}
// set_bit sets the bit at the given index to the given value.
pub fn (mut a Integer) set_bit(i u32, value bool) {
check_sign(a)
target_index := i / 32
offset := i % 32
if target_index >= a.digits.len {
if value {
a = one_int.lshift(i).bitwise_or(a)
}
return
}
mut copy := a.digits.clone()
if value {
copy[target_index] |= 1 << offset
} else {
copy[target_index] &= ~(1 << offset)
}
a = Integer{
signum: a.signum
digits: copy
}
}
// bitwise_or returns the "bitwise or" of the integers `a` and `b`.
pub fn (a Integer) bitwise_or(b Integer) Integer {
check_sign(a)
check_sign(b)
mut result := []u32{len: imax(a.digits.len, b.digits.len)}
bitwise_or_digit_array(a.digits, b.digits, mut result)
return Integer{
digits: result
signum: if result.len == 0 { 0 } else { 1 }
}
}
// bitwise_and returns the "bitwise and" of the integers `a` and `b`.
pub fn (a Integer) bitwise_and(b Integer) Integer {
check_sign(a)
check_sign(b)
mut result := []u32{len: imax(a.digits.len, b.digits.len)}
bitwise_and_digit_array(a.digits, b.digits, mut result)
return Integer{
digits: result
signum: if result.len == 0 { 0 } else { 1 }
}
}
// bitwise_not returns the "bitwise not" of the integer `a`.
pub fn (a Integer) bitwise_not() Integer {
check_sign(a)
mut result := []u32{len: a.digits.len}
bitwise_not_digit_array(a.digits, mut result)
return Integer{
digits: result
signum: if result.len == 0 { 0 } else { 1 }
}
}
// bitwise_xor returns the "bitwise exclusive or" of the integers `a` and `b`.
pub fn (a Integer) bitwise_xor(b Integer) Integer {
check_sign(a)
check_sign(b)
mut result := []u32{len: imax(a.digits.len, b.digits.len)}
bitwise_xor_digit_array(a.digits, b.digits, mut result)
return Integer{
digits: result
signum: if result.len == 0 { 0 } else { 1 }
}
}
// lshift returns the integer `a` shifted left by `amount` bits.
[direct_array_access]
pub fn (a Integer) lshift(amount u32) Integer {
if a.signum == 0 {
return a
}
if amount == 0 {
return a
}
normalised_amount := amount & 31
digit_offset := int(amount >> 5)
mut new_array := []u32{len: a.digits.len + digit_offset}
for index in 0 .. a.digits.len {
new_array[index + digit_offset] = a.digits[index]
}
if normalised_amount > 0 {
shift_digits_left(new_array, normalised_amount, mut new_array)
}
return Integer{
digits: new_array
signum: a.signum
}
}
// rshift returns the integer `a` shifted right by `amount` bits.
[direct_array_access]
pub fn (a Integer) rshift(amount u32) Integer {
if a.signum == 0 {
return a
}
if amount == 0 {
return a
}
normalised_amount := amount & 31
digit_offset := int(amount >> 5)
if digit_offset >= a.digits.len {
return zero_int
}
mut new_array := []u32{len: a.digits.len - digit_offset}
for index in 0 .. new_array.len {
new_array[index] = a.digits[index + digit_offset]
}
if normalised_amount > 0 {
shift_digits_right(new_array, normalised_amount, mut new_array)
}
return Integer{
digits: new_array
signum: a.signum
}
}
// binary_str returns the binary string representation of the integer `a`.
[direct_array_access]
pub fn (integer Integer) binary_str() string {
// We have the zero integer
if integer.signum == 0 {
return '0'
}
// Add the sign if present
sign_needed := integer.signum == -1
mut result_builder := strings.new_builder(integer.bit_len() + if sign_needed { 1 } else { 0 })
if sign_needed {
result_builder.write_string('-')
}
result_builder.write_string(u32_to_binary_without_lz(integer.digits[integer.digits.len - 1]))
for index := integer.digits.len - 2; index >= 0; index-- {
result_builder.write_string(u32_to_binary_with_lz(integer.digits[index]))
}
return result_builder.str()
}
// hex returns the hexadecimal string representation of the integer `a`.
[direct_array_access]
pub fn (integer Integer) hex() string {
// We have the zero integer
if integer.signum == 0 {
return '0'
}
// Add the sign if present
sign_needed := integer.signum == -1
mut result_builder := strings.new_builder(integer.digits.len * 8 +
if sign_needed { 1 } else { 0 })
if sign_needed {
result_builder.write_string('-')
}
result_builder.write_string(u32_to_hex_without_lz(integer.digits[integer.digits.len - 1]))
for index := integer.digits.len - 2; index >= 0; index-- {
result_builder.write_string(u32_to_hex_with_lz(integer.digits[index]))
}
return result_builder.str()
}
// radix_str returns the string representation of the integer `a` in the specified radix.
pub fn (integer Integer) radix_str(radix u32) string {
if integer.signum == 0 {
return '0'
}
return match radix {
2 {
integer.binary_str()
}
16 {
integer.hex()
}
else {
integer.general_radix_str(radix)
}
}
}
fn (integer Integer) general_radix_str(radix u32) string {
divisor := integer_from_u32(radix)
mut current := integer.abs()
mut new_current := zero_int
mut digit := zero_int
mut rune_array := []rune{cap: current.digits.len * 4}
for current.signum > 0 {
new_current, digit = current.div_mod(divisor)
rune_array << big.digit_array[digit.int()]
unsafe { digit.free() }
unsafe { current.free() }
current = new_current
}
if integer.signum == -1 {
rune_array << `-`
}
rune_array.reverse_in_place()
return rune_array.string()
}
// str returns the decimal string representation of the integer `a`.
pub fn (integer Integer) str() string {
return integer.radix_str(10)
}
fn u32_to_binary_without_lz(value u32) string {
return strconv.format_uint(value, 2)
}
fn u32_to_binary_with_lz(value u32) string {
mut result_builder := strings.new_builder(32)
binary_result := strconv.format_uint(value, 2)
result_builder.write_string(strings.repeat(`0`, 32 - binary_result.len))
result_builder.write_string(binary_result)
return result_builder.str()
}
fn u32_to_hex_without_lz(value u32) string {
return strconv.format_uint(value, 16)
}
fn u32_to_hex_with_lz(value u32) string {
mut result_builder := strings.new_builder(8)
hex_result := strconv.format_uint(value, 16)
result_builder.write_string(strings.repeat(`0`, 8 - hex_result.len))
result_builder.write_string(hex_result)
return result_builder.str()
}
// int returns the integer value of the integer `a`.
// NOTE: This may cause loss of precision.
pub fn (a Integer) int() int {
if a.signum == 0 {
return 0
}
// Check for minimum value int
if a.digits[0] == 2147483648 && a.signum == -1 {
return -2147483648
}
// Rest of the values should be fine
value := int(a.digits[0] & 0x7fffffff)
return value * a.signum
}
// bytes returns the a byte representation of the integer a, along with the signum int.
// NOTE: The byte array returned is in big endian order.
[direct_array_access]
pub fn (a Integer) bytes() ([]u8, int) {
if a.signum == 0 {
return []u8{len: 0}, 0
}
mut result := []u8{cap: a.digits.len * 4}
mut mask := u32(0xff000000)
mut offset := 24
mut non_zero_found := false
for index := a.digits.len - 1; index >= 0; {
value := u8((a.digits[index] & mask) >> offset)
non_zero_found = non_zero_found || value != 0
if non_zero_found {
result << value
}
mask >>= 8
offset -= 8
if offset < 0 {
mask = u32(0xff000000)
offset = 24
index--
}
}
return result, a.signum
}
// factorial returns the factorial of the integer `a`.
pub fn (a Integer) factorial() Integer {
if a.signum == 0 {
return one_int
}
mut product := one_int
mut current := a
for current.signum != 0 {
product *= current
current.dec()
}
return product
}
// isqrt returns the closest integer square root of the given integer.
pub fn (a Integer) isqrt() Integer {
if a.signum < 0 {
panic('Cannot obtain square root of negative integer')
}
if a.signum == 0 {
return a
}
if a.digits.len == 1 && a.digits.last() == 1 {
return a
}
mut shift := a.bit_len()
if shift & 1 == 1 {
shift += 1
}
mut result := zero_int
for shift >= 0 {
result = result.lshift(1)
larger := result + one_int
if (larger * larger).abs_cmp(a.rshift(u32(shift))) <= 0 {
result = larger
}
shift -= 2
}
return result
}
[inline]
fn bi_min(a Integer, b Integer) Integer {
return if a < b { a } else { b }
}
[inline]
fn bi_max(a Integer, b Integer) Integer {
return if a > b { a } else { b }
}
// gcd returns the greatest common divisor of the two integers `a` and `b`.
pub fn (a Integer) gcd(b Integer) Integer {
if a.signum == 0 {
return b.abs()
}
if b.signum == 0 {
return a.abs()
}
if a.abs_cmp(one_int) == 0 || b.abs_cmp(one_int) == 0 {
return one_int
}
return gcd_binary(a.abs(), b.abs())
}
// Inspired by the 2013-christmas-special by D. Lemire & R. Corderoy https://en.algorithmica.org/hpc/analyzing-performance/gcd/
// For more information, refer to the Wikipedia article: https://en.wikipedia.org/wiki/Binary_GCD_algorithm
// Discussion and further information: https://lemire.me/blog/2013/12/26/fastest-way-to-compute-the-greatest-common-divisor/
fn gcd_binary(x Integer, y Integer) Integer {
mut a, az := x.rsh_to_set_bit()
mut b, bz := y.rsh_to_set_bit()
shift := umin(az, bz)
for a.signum != 0 {
diff := b - a
b = bi_min(a, b)
a, _ = diff.abs().rsh_to_set_bit()
}
return b.lshift(shift)
}
// mod_inverse calculates the multiplicative inverse of the integer `a` in the ring `/n`.
// Therefore, the return value `x` satisfies `a * x == 1 (mod m)`.
// An error is returned if `a` and `n` are not relatively prime, i.e. `gcd(a, n) != 1` or
// if n <= 1
[inline]
pub fn (a Integer) mod_inverse(n Integer) !Integer {
return if n.bit_len() <= 1 {
error('math.big: Modulus `n` must be greater than 1')
} else if a.gcd(n) != one_int {
error('math.big: No multiplicative inverse')
} else {
a.mod_inv(n)
}
}
// this is an internal function, therefore we assume valid inputs,
// i.e. m > 1 and gcd(a, m) = 1
// see pub fn mod_inverse for details on the result
// -----
// the algorithm is based on the Extended Euclidean algorithm which computes `ax + by = d`
// in this case `b` is the input integer `a` and `a` is the input modulus `m`. The extended
// Euclidean algorithm calculates the greatest common divisor `d` and two coefficients `x` and `y`
// satisfying the above equality.
//
// For the sake of clarity, we refer to the input integer `a` as `b` and the integer `m` as `a`.
// If `gcd(a, b) = d = 1` then the coefficient `y` is known to be the multiplicative inverse of
// `b` in ring `Z/aZ`, since reducing `ax + by = 1` by `a` yields `by == 1 (mod a)`.
[direct_array_access]
fn (a Integer) mod_inv(m Integer) Integer {
mut n := Integer{
digits: m.digits.clone()
signum: 1
}
mut b := a
mut x := one_int
mut y := zero_int
if b.signum < 0 || b.abs_cmp(n) >= 0 {
b = b % n
}
mut sign := -1
for b != zero_int {
q, r := if n.bit_len() == b.bit_len() {
one_int, n - b
} else {
n.div_mod(b)
}
n = b
b = r
// tmp := q * x + y
tmp := if q == one_int {
x
} else if q.digits.len == 1 && q.digits[0] & (q.digits[0] - 1) == 0 {
x.lshift(u32(bits.trailing_zeros_32(q.digits[0])))
} else {
q * x
} + y
y = x
x = tmp
sign = -sign
}
if sign < 0 {
y = m - y
}
$if debug {
assert n == one_int
}
return if y.signum > 0 && y.abs_cmp(m) < 0 {
y
} else {
y % m
}
}
// rsh_to_set_bit returns the integer `x` shifted right until it is odd and an exponent satisfying
// `x = x1 * 2^n`
// we don't return `2^n`, because the caller may be able to use `n` without allocating an Integer
[direct_array_access; inline]
fn (x Integer) rsh_to_set_bit() (Integer, u32) {
if x.digits.len == 0 {
return zero_int, 0
}
mut n := u32(0)
for x.digits[n] == 0 {
n++
}
n = (n << 5) + u32(bits.trailing_zeros_32(x.digits[n]))
return x.rshift(n), n
}
[direct_array_access; inline]
fn (x Integer) is_odd() bool {
return x.digits[0] & 1 == 1
}
// is_power_of_2 returns true when the integer `x` satisfies `2^n`, where `n >= 0`
[inline]
pub fn (x Integer) is_power_of_2() bool {
return x.bitwise_and(x - one_int).bit_len() == 0
}
// bit_len returns the number of bits required to represent the integer `a`.
[inline]
pub fn (x Integer) bit_len() int {
if x.signum == 0 {
return 0
}
if x.digits.len == 0 {
return 0
}
return x.digits.len * 32 - bits.leading_zeros_32(x.digits.last())
}