```python
def mask_from_prefix(prefix):
    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):
    if not isinstance(mask, str):
        raise ValueError("mask must be a string")
    parts = mask.split(".")
    if len(parts) != 4 or any(part not.isdigit() for part in parts) or any(int(part) > 255 for part in parts):
        raise ValueError("Invalid mask format")
    prefix = 0
    for part in parts:
        prefix = prefix * 256 + int(part)
    return prefix
```