```python
def mask_from_prefix(prefix: int) -> str:
    if isinstance(prefix, bool) or not isinstance(prefix, int) or not 0 <= prefix <= 32:
        raise ValueError("prefix must be an integer from 0 to 32")
    octets = []
    for slot in range(4):
        ones = min(8, max(0, prefix - slot * 8))
        octets.append(str(256 - 2 ** (8 - ones)))
    return ".".join(octets)


def prefix_from_mask(mask: str) -> int:
    if not isinstance(mask, str):
        raise ValueError("mask must be a string")
    fields = mask.split('.')
    if len(fields) != 4:
        raise ValueError("mask must contain exactly four dot-separated fields")
    ones = 0
    for field in fields:
        if not field.isdigit():
            raise ValueError("field must contain only decimal digits")
        if field.startswith('0') and field[1:] != '':
            raise ValueError("field with leading zero other than 0 itself")
        if int(field) > 255:
            raise ValueError("octet above 255")
        ones += int(field)
    if ones > 32:
        raise ValueError("mask whose one bits are broken by a zero")
    return ones
```