Hash function

Hash functions are quite useful, right? In the past, we delved into their uses and some of their applications in data storage, fast searching, password storage, and checksums. However, we have yet to unpack what a good hash function actually looks like. In this topic, we'll explore the characteristics of an effective hash function, and examine some examples that fulfill these requirements and others that deliberately disregard them for distinct reasons.

Identifying a good hash function

Before we get started, let's refresh our understanding of hash functions. Basically, they operate like standard functions: you feed in some data, and they return a different output.

To be more specific, they utilize mathematical transformations to convert variable-sized input data into a fixed-size output called the hash value or hash code. So, what distinguishes a good hash function from an average one? Quite shortly, a noteworthy hash function displays several basic traits. Here are the properties we're particularly interested in:

• Efficiency: An efficient hash function should perform data processing quickly, optimizing computational resources for actual applications.

• Determinism means the function will result in the same output for a particular input no matter when and how often it's applied.

• Uniformity: The hash values should be uniformly distributed, meaning the inputs should map evenly among possible hash values.

Eager to learn more? Hang tight! We'll dive deeper into each of these properties in the upcoming sections.

Efficiency

Efficiency is the lifeblood of practical hash functions, enabling fast data processing for applications like storage with rapid searching. Let's break this down:

Imagine you're working with a database system, like a vast library, that holds millions of books (records). Each book has a unique code (unique identifier). Now, consider a hash function as a super-efficient librarian who can instantly tell you exactly where a book is located when given its unique code. This librarian (hash function) quickly transforms this code into a location (hash value), enabling you to find your book swiftly. This is the beauty of an efficient hash function — it allows for rapid data retrieval and searching.

As we will learn in the following topics, the speed at which a hash function can compute hash values is typically constant, denoted as $O(1)$. This speed is vital for efficient searching, and most hash tables meet this criterion.

However, there are exceptions. Some cryptographic hash functions, such as SHA-1, may take linear time $O(n)$ to compute a hash value because they process input data in chunks. However, the complexity is not always strictly $O(n)$ — it depends on the input length and specific implementation. This is different from hash functions used in hash tables, which typically aim for $O(1)$ complexity.

This is not necessarily a downside: in fact, it's perfectly acceptable for certain tasks, such as creating checksums and hashing passwords. The point is, the desired 'speed' of a hash function can vary depending on the specific use case it's applied to.

Determinism

The concept of determinism in hash functions is a vital aspect to understand in computer science, not only in hash functions. To simplify it, if you have two identical inputs, they should generate the same hash value.

In essence, a deterministic function is one that is not random. To illustrate, consider a function that randomly returns either 0 or 1, irrespective of the input. While this is technically a hash function, it's not a deterministic one because the output is not consistent for the same input. Indeed, it is possible that the first time you calculate the hash value of your input $n$, you get $h(n) = 0$, and the second time you get $h(n) = 1$, which is absurd.

Anyway, can nondeterministic hash functions exist at all? It seems a bit unthinkable, doesn't it? Let's imagine you have two separate variables, and both carry the same value of 7. From a computer's perspective, these variables are distinct because they occupy different positions in its memory. But when you compare the values, they are identical. In such a scenario, the ideal hash function should return the same output. If a hash function returns the memory address of the value instead, it doesn't meet the deterministic condition. Similar functions can be useful on very specific situations.

Importance of determinism

Now, why is determinism important? It guarantees consistent results for a given set of inputs. This is particularly crucial for password storage. A hash function is employed to convert user passwords into hash values. If the hash function is deterministic, it ensures that each time a user types in their password, it yields the same hash value. This hash value is then used for comparison during the authentication process. Without deterministic hash functions, the process of password authentication would turn chaotic, making it nearly impossible to verify user identities.

Imagine a scenario where a user enters their password, and it generates a different hash value each time. The system would not be able to authenticate the user, leading to a breakdown in security protocols. Hence, the concept of determinism in hash functions is not just a theoretical aspect of computer science, but a practical necessity in maintaining secure systems.

Uniformity

Uniformity is another crucial attribute of hash functions. This essentially means that the hash function should spread out its output (hash values) as evenly as possible. Let's consider it like this — if we categorize all possible outputs from a hash function, we'd like each category or group to be roughly the same size. This helps to avoid a bottleneck or slow-down caused by too many outputs (collisions) clustering around one or a few values, known as peak keys.

In the field of cryptography, if certain groups of passwords or messages experience many collisions, they become more susceptible to hacking attempts. This could pose similar issues in systems dealing with checksums or data storage.

For instance, imagine a data storage system using a hash function that generates hash values based on the last digit of a user's ID. If the majority of these IDs end in even digits, the hash values will tend to pile up in certain 'buckets' or groups, creating uneven distribution. This clustering leads to a higher risk of collisions, which can hinder the system's overall performance and raise the likelihood of data errors and retrieval problems.

In other words, this lack of uniformity in the hash function can negatively impact the reliability and efficiency of the system.

Conclusion

By now, you're aware of what hash functions are and what constitutes a good one. We dissected the typical properties of suitable hash functions, specifically:

• Efficiency

• Determinism

• Uniformity.

Now you're equipped to probe further, familiarize yourself with typical and cryptographic hash functions, and study other hashing techniques like hash tables!