Choosing a prime number for the hash table size in the division method helps ensure a more uniform distribution of hash values and reduces clustering caused by patterns in the key values. When the table size is a prime, it avoids situations where certain numerical patterns in the keys (such as keys that are multiples of a common factor) consistently hash to the same subset of indices. This is because primes have fewer divisors, making it less likely for the modulus operation to produce repeated results due to regularities in the input keys. For example, if the table size is 10 and most keys are even numbers, they’ll always hash to even indices (e.g. 0, 2, 4, 6, 8). But if the table size is 7 (a prime), the modulus operation spreads values more evenly. This principle improves performance by minimising collisions and helping the hash function approximate true randomness across the index range.

The type of key used has a significant impact on how a hash function is designed. With numeric keys, the process is generally simpler because arithmetic operations like division or modulus can be applied directly. However, with string keys, additional steps are required to convert the characters into numeric form before a hash value can be computed. This typically involves mapping each character to its ASCII (or Unicode) code and combining these values through methods like summation, positional weighting, or polynomial rolling. The length of the string, character set, and expected variations in input should all be considered. For example, short strings like "ab" and "ba" have the same characters but may need to hash to different values to avoid collisions. A good string hash function also avoids simple summing because it can produce identical results for anagrams. Therefore, the key type determines whether the hash function prioritises simplicity or complexity to maintain speed, uniqueness, and consistency.

A poorly designed hash function significantly degrades the performance of a hash table. If the function does not distribute keys uniformly, it leads to clustering, where many keys are stored at or near the same index. This increases the time required to resolve collisions, especially in methods like linear probing, where each additional collision adds to the sequence of comparisons. If the function has a high collision rate, more time is spent handling multiple entries at a single index through techniques like chaining or rehashing, reducing the average-case efficiency from constant time (O(1)) to linear time (O(n)). Additionally, if the function is computationally expensive, even with good distribution, the overhead of calculating the hash value diminishes the speed advantage of using a hash table. Inconsistent or non-deterministic functions can also lead to data being irretrievable, as the same key might not point to the same index during a lookup. Hence, good hash function design is critical for optimal performance.

Yes, even in a well-designed hash function, two completely different keys can produce the same hash value. This event is known as a collision. Collisions are a natural consequence of using a finite-sized hash table with a larger or infinite set of possible keys. Because there are more potential keys than there are indices in the table, it is mathematically inevitable that some keys will map to the same location. Even with an ideal hash function that ensures excellent distribution and low collision rates, you cannot eliminate the possibility of collisions entirely. The key difference with well-designed hash functions is that they make collisions infrequent and unpredictable, which ensures the performance of the hash table remains consistently high. Effective collision resolution strategies, such as chaining or open addressing with probing, are always needed regardless of how good the hash function is. Therefore, developers must plan for collisions in all practical hash table implementations, even when using robust hash functions.

When a hash table is dynamically resized—typically due to reaching a high load factor—the hash function must adapt to the new table size. This adaptation is necessary because the original hash values were calculated using the old table size (usually in the modulus step), and changing the size alters the result of the hash function. For example, if the hash function used h(k) = k mod 10 in a table of size 10, and the table is resized to 20, the same key will now hash to a different index using h(k) = k mod 20. Therefore, during resizing, all existing key-value pairs must be rehashed using the new hash function that incorporates the updated table size. This means each key is recalculated and inserted into the appropriate position in the new table. This process ensures that data remains correctly positioned for efficient access, but it is computationally expensive. Hence, resizing is typically performed infrequently and only when performance would otherwise be severely impacted by collisions.