# Unsigned 64-bit Integer Operations

Module `std::math::u64`

contains a set of procedures which can be used to perform unsigned 64-bit integer operations. These operations fall into the following categories:

**Arithmetic operations**- addition, multiplication, division etc.**Comparison operations**- equality, less than, greater than etc.**Bitwise operations**- binary AND, OR, XOR, bit shifts etc.

All procedures assume that an unsigned 64-bit integer (u64) is encoded using two elements, each containing an unsigned 32-bit integer (u32). When placed on the stack, the least-significant limb is assumed to be deeper in the stack. For example, a u64 value `a`

consisting of limbs `a_hi`

and `a_lo`

would be position on the stack like so:

`[a_hi, a_lo, ... ]`

Procedures which check whether the input values are encoded correctly are designated with `checked`

prefix. For example, `checked_add`

would fail if any of the top 4 elements on the stack contains a value greater than $2^{32} - 1$. In contrast, `wrapping_add`

and `overflowing_add`

would not perform these checks, and therefore, if any of the top 4 stack elements is greater than $2^{32} - 1$, the operation will not fail but rather will produce an undefined result. Thus, when using versions of procedures which are not checked, it is important to be certain that input values are 32-bit limbs encoding valid u64 values.

## Arithmetic operations

Procedure | Description |
---|---|

checked_add | Performs addition of two unsigned 64-bit integers and fails if the result would overflow. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a + b) % 2^64 |

overflowing_add | Performs addition of two unsigned 64-bit integers preserving the overflow. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [overflow_flag, c_hi, c_lo, ...], where c = (a + b) % 2^64 |

wrapping_add | Performs addition of two unsigned 64-bit integers discarding the overflow. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a + b) % 2^64 |

checked_sub | Performs subtraction of two unsigned 64-bit integers and fails if the result would underflow. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a - b) % 2^64 |

overflowing_sub | Performs subtraction of two unsigned 64-bit integers preserving the overflow. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [underflow_flag, c_hi, c_lo, ...], where c = (a - b) % 2^64 |

wrapping_sub | Performs subtraction of two unsigned 64-bit integers discarding the overflow. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a - b) % 2^64 |

checked_mul | Performs multiplication of two unsigned 64-bit integers and fails if the result would overflow. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a * b) % 2^64 |

overflowing_mul | Performs multiplication of two unsigned 64-bit integers preserving the overflow. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi_hi, c_hi_lo, c_lo_hi, c_lo_lo, ...], where c = (a * b) % 2^64 |

wrapping_mul | Performs multiplication of two unsigned 64-bit integers discarding the overflow. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = (a * b) % 2^64 |

checked_div | Performs division of two unsigned 64-bit integers discarding the remainder. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a // b |

unchecked_div | Performs division of two unsigned 64-bit integers discarding the remainder. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a // b |

checked_mod | Performs modulo operation of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a % b |

unchecked_mod | Performs modulo operation of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a % b |

checked_divmod | Performs divmod operation of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [r_hi, r_lo, q_hi, q_lo ...], where r = a % b, q = a // b |

unchecked_divmod | Performs divmod operation of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [r_hi, r_lo, q_hi, q_lo ...], where r = a % b, q = a // b |

## Comparison operations

Procedure | Description |
---|---|

checked_lt | Performs less-than comparison of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a < b, and 0 otherwise. |

unchecked_lt | Performs less-than comparison of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a < b, and 0 otherwise. |

checked_gt | Performs greater-than comparison of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a > b, and 0 otherwise. |

unchecked_gt | Performs greater-than comparison of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a > b, and 0 otherwise. This takes 11 cycles. |

checked_lte | Performs less-than-or-equal comparison of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a <= b, and 0 otherwise. |

unchecked_lte | Performs less-than-or-equal comparison of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a <= b, and 0 otherwise. |

checked_gte | Performs greater-than-or-equal comparison of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a >= b, and 0 otherwise. |

unchecked_gte | Performs greater-than-or-equal comparison of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a >= b, and 0 otherwise. |

checked_eq | Performs equality comparison of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a == b, and 0 otherwise. |

unchecked_eq | Performs equality comparison of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a == b, and 0 otherwise. |

checked_neq | Performs inequality comparison of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a != b, and 0 otherwise. |

unchecked_neq | Performs inequality comparison of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c, ...], where c = 1 when a != b, and 0 otherwise. |

checked_eqz | Performs comparison to zero of an unsigned 64-bit integer. The input value is assumed to be represented using 32-bit limbs, fails if it is not. The stack transition looks as follows: [a_hi, a_lo, ...] -> [c, ...], where c = 1 when a == 0, and 0 otherwise. |

unchecked_eqz | Performs comparison to zero of an unsigned 64-bit integer. The input value is assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [a_hi, a_lo, ...] -> [c, ...], where c = 1 when a == 0, and 0 otherwise. |

checked_min | Compares two unsigned 64-bit integers and drop the larger one from the stack. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a when a < b, and b otherwise. |

unchecked_min | Compares two unsigned 64-bit integers and drop the larger one from the stack. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a when a < b, and b otherwise. |

checked_max | Compares two unsigned 64-bit integers and drop the smaller one from the stack. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a when a > b, and b otherwise. |

unchecked_max | Compares two unsigned 64-bit integers and drop the smaller one from the stack. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a when a > b, and b otherwise. |

## Bitwise operations

Procedure | Description |
---|---|

checked_and | Performs bitwise AND of two unsigned 64-bit integers. The input values are assumed to be represented using 32-bit limbs, but this is not checked. The stack transition looks as follows: [[b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a AND b. |

checked_or | Performs bitwise OR of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a OR b. |

checked_xor | Performs bitwise XOR of two unsigned 64-bit integers. The input values are expected to be represented using 32-bit limbs, and the procedure will fail if they are not. The stack transition looks as follows: [b_hi, b_lo, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a XOR b. |

overflowing_shl | Performs left shift of one unsigned 64-bit integer preserving the overflow and using the pow2 operation. The input value to be shifted is assumed to be represented using 32-bit limbs. The shift value should be in the range [0, 64), otherwise it will result in an error. The stack transition looks as follows: [b, a_hi, a_lo, ...] -> [d_hi, d_lo, c_hi, c_lo, ...], where (d,c) = a << b, which d contains the bits shifted out. This takes 35 cycles. |

unchecked_shl | Performs left shift of one unsigned 64-bit integer using the pow2 operation. The input value to be shifted is assumed to be represented using 32-bit limbs. The shift value should be in the range [0, 64), otherwise it will result in an error. The stack transition looks as follows: [b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a << b mod 2^64. This takes 28 cycles. |

overflowing_shr | Performs right shift of one unsigned 64-bit integer preserving the overflow and using the pow2 operation. The input value to be shifted is assumed to be represented using 32-bit limbs. The shift value should be in the range [0, 64), otherwise it will result in an error. The stack transition looks as follows: [b, a_hi, a_lo, ...] -> [d_hi, d_lo, c_hi, c_lo, ...], where c = a >> b, d = a << (64 - b). This takes 94 cycles. |

unchecked_shr | Performs right shift of one unsigned 64-bit integer using the pow2 operation. The input value to be shifted is assumed to be represented using 32-bit limbs. The shift value should be in the range [0, 64), otherwise it will result in an error. The stack transition looks as follows: [b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a >> b. This takes 44 cycles. |

unchecked_rotl | Performs left rotation of one unsigned 64-bit integer using the pow2 operation. The input value to be shifted is assumed to be represented using 32-bit limbs. The shift value should be in the range [0, 64), otherwise it will result in an error. The stack transition looks as follows: [b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a << b mod 2^64. This takes 35 cycles. |

unchecked_rotr | Performs right rotation of one unsigned 64-bit integer using the pow2 operation. The input value to be shifted is assumed to be represented using 32-bit limbs. The shift value should be in the range [0, 64), otherwise it will result in an error. The stack transition looks as follows: [b, a_hi, a_lo, ...] -> [c_hi, c_lo, ...], where c = a << b mod 2^64. This takes 40 cycles. |