Hash functions play a crucial role in various security applications, including password storage (hash values instead of passwords), digital signatures, and data integrity checks. Hash values, or message digests, are values that a hash function returns. The hash function is shown in the image below:

Properties of hash functions

Pre-Image Resistance

A cryptographic hash function h is preimage resistant if it is computationally infeasible to determine the original input x given only the hash output h(x). This means that even if h(x) is known, finding an x such that h(x) produces the given output is extremely difficult.

Second Pre-Image Resistance

A cryptographic hash function h satisfies second preimage resistance if:

• Given an input x₁ and its hash h(x₁), it should be computationally infeasible to find another input x₂ such that h(x₁) = h(x₂).
• In other words, an attacker should not be able to find a different input x₂ that produces the same hash as x₁.
• This property ensures the integrity of digital signatures and stored passwords by preventing intentional collisions.
• Second preimage resistance is crucial in preventing forgery and tampering with hashed data.

Collision Resistance

A cryptographic hash function h satisfies collision resistance if:

• It is computationally infeasible to find two different inputs x₁ and x₂ such that h(x₁) = h(x₂).
• In other words, no two distinct inputs should produce the same hash value.
• This property ensures the security of digital signatures, certificates, and cryptographic integrity checks by preventing attackers from generating fraudulent data with the same hash.
• Collision resistance is crucial in maintaining the uniqueness of hashes and protecting against deliberate hash collisions.