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 } // pow returns the integer `a` raised to the power of the u32 `exponent`. pub fn (a Integer) pow(exponent u32) Integer { if exponent == 0 { return one_int } if exponent == 1 { return a.clone() } mut n := exponent mut x := a 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 `divisor`. pub fn (a Integer) mod_pow(exponent u32, divisor Integer) Integer { if exponent == 0 { return one_int } if exponent == 1 { return a % divisor } mut n := exponent mut x := a % divisor mut y := one_int for n > 1 { if n & 1 == 1 { y *= x % divisor } x *= x % divisor n >>= 1 } return x * y % divisor } // big_mod_power returns the integer `a` raised to the power of the integer `exponent` modulo the integer `divisor`. [direct_array_access] pub fn (a Integer) big_mod_pow(exponent Integer, divisor Integer) Integer { if exponent.signum < 0 { panic('Exponent needs to be non-negative.') } if exponent.signum == 0 { return one_int } mut x := a % divisor mut y := one_int mut n := u32(0) // For all but the last digit of the exponent for index in 0 .. exponent.digits.len - 1 { n = exponent.digits[index] for _ in 0 .. 32 { if n & 1 == 1 { y *= x % divisor } x *= x % divisor n >>= 1 } } // Last digit of the exponent n = exponent.digits.last() for n > 1 { if n & 1 == 1 { y *= x % divisor } x *= x % divisor n >>= 1 } return x * y % divisor } // 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 } } [direct_array_access] fn (bi Integer) msb() u32 { for idx := 0; idx < bi.digits.len; idx += 1 { word := bi.digits[idx] if word > 0 { return u32((idx * 32) + bits.trailing_zeros_32(word)) } } return u32(32) } // 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.signum < 0 { return a.neg().gcd(b) } if b.signum < 0 { return a.gcd(b.neg()) } if a.digits.len + b.digits.len <= 2 { return gcd_euclid(a, b) } else { return gcd_binary(a, b) } } fn gcd_euclid(x Integer, y Integer) Integer { mut a := x mut b := y mut r := a % b for r.signum != 0 { a = b b = r r = a % b } return b } // 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 := x mut b := y mut az := a.msb() bz := b.msb() shift := umin(az, bz) b = b.rshift(bz) for a.signum != 0 { a = a.rshift(az) diff := b - a az = diff.msb() b = bi_min(a, b) a = diff.abs() } return b.lshift(shift) } // 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()) }