*/ secure hash function and relate my attempts to come up with a "toy" ... A Good Hash Function is Hard to Find,and Vice Versa This is a really long string of text which is going toJoshua Holden be the input to our hash function.Rose-Hulman Institute ofTechnology 01100011 ... Our first example doesn’t stack up too well. x &\gets x + \text{ROL}_k(x) \\ Now hash the string "gob". x &\gets px \\ Well, if I flip a high bit, it won't affect the lower bits because you can see multiplication as a form of overlay: Flipping a single bit will only change the integer forward, never backwards, hence it forms this blind spot. They're Technically, any function that maps all possible key values to a slot in the hash table is a hash function. x &\gets x \oplus (x \gg z) \\ }, /* This algorithm was created for the sdbm (a reimplementation of ndbm) It has several properties that distinguish it from the non-cryptographic one. Slight variations in the string should result in different hash Clearly there is some form of bias. while ( *name ) { The following are important properties that a cryptography-viable hash function needs to function properly: hash function. The difficult task is coming up with a good compression function. for a large input you would see certain statistical properties bad for a Many relatively simple components can be combined into a strong and robust non-cryptographic hash function for use in hash tables and in checksumming. Rule 1: Satisfies. Hash functions help to limit the range of the keys to the boundaries of the array, so we need a function that converts a large key into a smaller key. Without such hybrid, the behavior tends to be relatively local and not interfering well with each other. Let’s break it down step-by-step. constructing a hash function. As mentioned briefly in the previous section, there are multiple ways for \end{align*}\]. If you want good performance, you shouldn't read only one byte at a time. It serves for combining the old state and the new input block (\(x\)). These are my notes on the design of hash functions. Fetching multiple blocks and sequentially (without dependency until last) running a round is something I've found to work well. x &\gets x + 1 \\ If your diffusion function is primarily based on bitwise operations, you should use the additive combinator function. Hash function ought to be as chaotic as possible. return (hash%101); /* 101 is prime */ That's good, but we're not quite there yet... And voilà, we now have a perfect bit independence: So our finalized version of an example diffusion is, \[\begin{align*} So, I've been needing a hash function for various purposes, lately. A secure compression function acts like a keyed hash function that takes only a single fixed input block size. A small change in the input should appear in the output as if it was a big change. Crypto hashes are however slower, and tend to generate larger codes (256 bits or more) Using them to implement a bucketing strategy for 100 servers would be over-engineering. Clearly, hello is more likely to be a word than ctyhbnkmaasrt, but the hash function must not be affected by this statistical redundancy. Every hash function must do that, including the bad ones. Instead of shifting left, we need to shift right, since multiplication only affects upwards: \[\begin{align*} A uniform hash function produces clustering near 1.0 with high probability. In Bitcoin’s blockchain hashes are much more significant and are much more complicated because it uses one-way hash functions like SHA-256 which are very difficult to break. return sum % table_size; h = 0; We would like these data elements to still be distributable Rule 1: If something else besides the input data is used to determine the In a cryptographic hash function, it must be infeasible to: Non-cryptographic hash functions can be thought of as approximations of these invariants. */ unsigned int h, g; The ideal hash functions has the property that the distribution of image of a a subset of the domain is statistically independent of the probability of said subset occuring. h = ( h << 4 ) + *name++; */ \end{align*}\], (note that we have the \(+1\) in order to make it zero-sensitive), This generates following avalanche diagram. So let's take as an example the hash function used in the last section: Which rules does it break and satisfy? In this article, the author discusses the requirements for a secure hash function and relates his attempts to come up with a “toy” system which is both reasonably secure and also suitable for students to work with by hand in a classroom setting. x &\gets px \\ It is expected to have all the collision resistances that such a hash function would need. If the hash table size M is small compared to the resulting summations, then this hash function should do a good job of distributing strings evenly among the hash table slots, because it gives equal weight to all characters in the string. Use up and down arrows to review and enter to select. If \((x, y)\) is very red, the probability that \(d(a')\), where \(a'\) is \(a\) with the \(x\)'th bit flipped,' has the \(y\)'th bit flipped is very high. Multiple test suits for testing the quality and performance of your hash function. The next subdiffusion are of massive importance. { } In fact, if our hash function distributes any collisions evenly throughout the hash table, that means that we’ll never end up with one long linked list that’s bigger than everything else. x &\gets x \oplus (x \gg z) \\ char *p; unsigned long hash(char *name) That is, every hash value in the output range should be generated with roughly the same probability.The reason for this last requirement is that the cost of hashing-based methods goes up sharply as the number of collisions—pairs of inputs that are mapped to the same hash … Every hash function must do that, including Combining them is what creates a good diffusion function. a hash function quickly, djb2 is usually a good candidate as it is easily It is therefore important to differentiate between the algorithm and the function. An example of such combination function is simple addition. The key to a good hash function is to try-and-miss. data elements. Hash the string "bog". This has to do with the so-called instruction pipeline in which modern processors run instructions in parallel when they can. Hash functions convert a stream of arbitrary data bytes into a single number. So what makes for a good hash function? result, cutting down on the efficiency of the hash table. 4) The hash function generates very different hash values for similar strings. None of the existing hash functions I could find were sufficient for my needs, so I went and designed my own. In a sense, you can think of the ideal hash function as being a function where the output is uniformly distributed (e.g., chosen by a sequence of coinflips) over the codomain no matter what the distribution of the input is. Rule 3: Breaks. for (hash=0, i=0; i> 24; } h ^= g>>24; But not all hash functions are made the same, meaning different hash functions have different abilities. It takes in an input (often a string of characters) and returns a corresponding cryptographic "fingerprint" for that input (often another string of characters). Consider you have an english dictionary. Whenever you have a set of values where you want to be able to look up arbitrary elements quickly, a hash table is a good default data structure. As such, it is important to find a small, diverse set of subdiffusions which has a good quality. not so good in the long run. 1) The hash value is fully determined by the data being hashed. Smhasher is one of these. x &\gets x \oplus (x \gg z) \\ I present a new low-byte code based on base 3.…, LZ4 is an exciting algorithm, but unfortunately there is no good explanation on how it works. Just use a simple, fast, non-crypto algorithm for it. Rule 3: If the hash function does not uniformly distribute the data across Another similar often used subdiffusion in the same class is the XOR-shift: (note that \(m\) can be negative, in which case the bitshift becomes a right bitshift). A hash table is a great data structure for unordered sets of data. }, /* djb2 x &\gets x \oplus (x \gg z) \\ Avalanche diagrams are the best and quickist way to find out if your diffusion function has a good quality. h ^= g; It doesn't matter if the combinator function is commutative or not, but it is crucial that it is not biased, i.e. the entire set of possible hash values, a large number of collisions will One way to do that is to use some other well known cryptographic primitive. A good hash function should have the following properties: Efficiently computable. We will try to boil it down to few operations while preserving the quality of this diffusion. In its most general form, a hash function projects a value from a set with many members to a value from a set with a fixed number of members. The hash map data structure grows linearly to hold n elements for O(n) linear space complexity. Let's try multiplying by a prime: Now, this is quite interesting actually. In the random oracle model, instead of making a highly non-standard (and possibly unsubstantiated) assumption that “my system is secure with this H” (e.g., H being SHA-1), one proves that the system is at least secure with an “ideal” hash function H (under standard assumptions). char hash; int hashpjw(char *s) { 2.3.3 Hash. for( ; *str; str++) sum += *str; Now let me talk just very briefly about the particular hash function we're going to use. That's a pretty abstract description, so instead I like to imagine a hash function as a fingerprinting machine. Ideally, there should exist a bijection, \(g(f(a, b), b) = a\), which implies that it is not biased. The hash value is just the sum of all the input characters. }, /* Peter Weinberger's */ This operation usually returns the same hash for a given key. }, /* UNIX ELF hash There is an efficient test to detect most such weaknesses, and many functions pass this test. \end{align*}\]. Diffusions are often build by smaller, bijective components, which we will call "subdiffusions". x &\gets px \\ every input has one and only one output, and vice versa) hash functions, namely that input and output are uncorrelated: This diffusion function has a relatively small domain, for illustrational purpose. x &\gets x \oplus (x \gg z) \\ unsigned long hash(unsigned char *str) With a good hash function, it should be hard to distinguish between a truely random sequence and the hashes of some permutation of the domain. However, if a hash function is chosen well, then it is difficult to find two keys that will hash to the same value. Why is that? The most obvious think to remove is the rotation line. It's a good introductory example but That fingerprint is should be unique to that input, but if you were given some random fingerprint, you … And we're back again. It typically looks something like this: On the left we have m m m buckets. In this paper I will discuss the requirements for a secure hash function and relate my attempts to come up with a “toy ” system which both reasonably secure and also suitable for students to work with by hand in a classroom setting. This is called the hash function butterfly effect. The answer is pretty simple: shifting left moves the entropy upwards, hence the multiplication will never really flip the lower bits. This blog post tries to explain it in terms that everybody can understand.…. So how can we fix this (we don't want this bias)? But it hurts quality: Where do these blind spot comes from? return h; For coding up Hash function ought to be as chaotic as possible. web search will turn up hundreds) so we won't cover too many here except That is, collisions are not likely to occur even within non-uniform distributed sets. } values, but with this function they often don't. the same. So what do we do? If you are curious about how a hash function works, this Wikipedia article provides all the details about how the Secure Hash Algorithm 2 (SHA-2) works. Rule 2: Satisfies. Hash Functions Hash functions are an essential part of modern cryptographic practice. Another use of hashing: Rabin-Karp string searching. * database library and seems to work relatively well in scrambling bits 2) The hash function uses all the input data. Let's examine why each of these is important: What can cause these? By the pigeon-hole principle, many possible inputs will map to the same output. Crypto or non-crypto, every good hash function gives you a strong uniformity guarantee. input (often a string), and return s an integer in the range of possible hash, then the hash value is not as dependent upon the input data, thus of possible hash values. int hash(char *str, int table_size) However, some functions like bcrypt, which label themselves as password hash functions, define a maximum size input length (in the case of bcrypt, 72 bytes). { The hash function is a complex mathematical problem which the miners have to solve in order to find a block. if (g = h&0xF0000000) { A good hash function should map the expected inputs as evenly as possible over its output range. A small change in the input should appear in the output as if it was a big change. implemented and has relatively good statistical properties. The reason for the use of non-cryptographic hash function is that they're significantly faster than cryptographic hash functions. I get that is a somewhat good function to avoid collisions and a fast one, but how can I make a better one? In particular, we can eat \(N\) bytes of the input at once and modify the state based on that: \(f(s', x)\) is what we call our combinator function. Hash Functions Hash functions are an essential part of modern cryptographic practice. * many years ago in comp.lang.c If your diffusion isn't zero-sensitive (i.e., \(f(0) = \{0, 1\}\)), you should panic come up with something better. Rule 2: If the hash function doesn't use all the input data, then slight return hash; Remember that hash function takes the data as Rule 4: In real world applications, many data sets contain very similar Assuming a good hash function (one that minimizes collisions!) Two elements in the domain, \(a, b\) are said to collide if \(h(a) = h(b)\). We’ve established that a hash function can be thought of as a random oracle that, given some input x ∈ {0, 1} ∗ (i.e., an arbitrarily-sized sequence of bits) returns a “random,” fixed-size input y ∈ {0, 1}256 (i.e., 256 bits) and will always return that same y given that same x as input. To do that, we'll use a cryptographic hash function, also called a hashing algorithm, also called a Fancy McBuzzword Skidoo. Hash tables are used to implement map and set data structures in most common programming languages.In C++ and Java they are part of the standard libraries, while Python and Go have builtin dictionaries and maps.A hash table is an unordered collection of key-value pairs, where each key is unique.Hash tables offer a combination of efficient lookup, insert and delete operations.Neither arrays nor l… The second class is dependent bitwise subdiffusions. x &\gets px \\ This is the job of the hash function. So let’s see Bitcoin hash function, i.e., SHA-256 { If bucket i contains xi elements, then a good measure of clustering is (∑ i(xi2)/n) - α. int i; indices into the hash table. we usually have O(1) constant get/set complexity. x &\gets x + 1 \\ That's kind of boring, let's try adding a number: Meh, this is kind of obvious. This is where hash functions come in to play. Uniformity. This is an example of the folding approach to designing a hash function. What is a good hash function? int c; Difussions can be thought of as bijective (i.e. // Make sure a valid string passed in uniformly distribute the strings, but if you were to analyze this function // Return the sum mod the table size Indeed if you combining enough different subdiffusions, you get a good diffusion function, but there is a catch: The more subdiffusions you combine the slower it is to compute. over a hash table. A better function is considered the last three digits. A hash function is a function that deterministically maps an arbitrarily large input space into a fixed output space. Another virtue of a secure hash function is that its output is not easy to predict. For example, if we flip the sixth bit, and trace it down the operations, you will how it never flips in the other end. and turns it … As mentioned, a hashing algorithm is a program to apply the hash function to an input, according to several successive sequences whose number may vary according to the algorithms. If you are a programmer, you must have heard the term "hash function". So what makes for a good hash function? Generate two inputs with the same output. to present a few decent examples of hash functions: You get the idea... there are many possible hash functions. I gave code for the fastest such function I could find. In particular, make sure your diffusion contains at least one zero-sensitive subdiffusion as component. Hash functions also come with a not-so-nice side effect: ... Any good hash function can be used and you just use h ... consider using up-to 32 bits. allowing for a worse distribution of the hash values. 1 1. This is called the hash function butterfly effect. { { The first class to consider is the bitwise subdiffusions. hashed. Okay, so we've talked about three properties of hash functions and one application of each of those. Hany F. Atlam, Gary B. Wills, in Advances in Computers, 2019. while (c = *str++) hash = ((hash << 5) + hash) + c; // hash*33 + c The notion of hash function is used as a way to search for data in a database. I saw a lot of hash function and applications in my data structures courses in college, but I mostly got that it's pretty hard to make a good hash function. Turns out that this bias mostly originates in the lack of hybrid arithmetic/bitwise sub. }. h &= ~g; However, if our hash function does a good job of distributing elements throughout the hash table, then we’ll be okay. x &\gets x \oplus (x \ll z) \\ Every character is summed. A Small Change Has a Big Impact. unsigned long hash = 0; for(p=s; *p!='\0'; p++){ Here's what a cryptographic hash functions does: it takes an input (a file, a string of text, a number, a private key, etc.) \(d(a)\) is just our diffusion function. To achieve a good hashing mechanism, It is important to have a good hash function with the following basic requirements: Easy to compute: It should be easy to compute and must not become an algorithm in itself. Should uniformly distribute the keys (Each table position equally likely for each key) For example: For phone numbers, a bad hash function is to take the first three digits. There are lots of hash functions in existence, but this is the one bitcoin uses, and it's a pretty good … the bad ones. The hash value is fully determined by the data being if ( g = h & 0xF0000000 ) 2) The hash function uses all the input data. It's the class of linear subdiffusions similar to the LCG random number generator: \[d(x) \equiv ax + c \pmod m, \quad \gcd(x, m) = 1\], (\(\gcd\) means "greatest common divisor", this constraint is necessary in order to have \(a\) have an inverse in the ring). Hash functions are functions which maps a infinite domain to a finite codomain. There are many possible ways to construct a better hash function (doing a hash values resulting in too many collisions. x &\gets x \oplus (x \gg z) \\ By reading multiple bytes at a time, your algorithm becomes several times faster. fact secure when instantiated with a “good” hash function. There are four main characteristics of a good hash function: Deriving such a function is really just coming up with the components to construct this hash function. x &\gets x + 1 \\ hash functions In general, hash functions take an input of any size and return an output of a … One must distinguish between the different kinds of subdiffusions. Breaking the problem down into small subproblems significantly simplifies analysis and guarantees. A common weakness in hash function is for a small set of input bits to cancel each other out. We call all the black area "blind spots", and you can see here that anything with \(x > y\) is a blind spot. if (str==NULL) return -1; }, char XORhash( char *key, int len) This time with two less instructions. A hash table is a large list of pre-computed hashes for commonly used passwords. * This algorithm was first reported by Dan Bernstein Value is just the sum of all the collision resistances that such a hash table you delve... Function has a good job of distributing elements throughout the hash function we 're to... Functions hash functions I could find output range one must distinguish between different... ( f ( a, b\ ) are uniformly distributed variables, \ ( f (,... Function should map the expected inputs as evenly as possible data in a cryptographic hash functions are essential. Values, but it is expected to have all the input should in... Has to do that, including the bad ones function ought to be as chaotic possible!: Efficiently computable it break and satisfy approach to designing a hash function ought to be as as... ( \ ( d ( a ) \ ) is too abstract description, so I. To solve in order to find a block then a good measure of is! Distributed variables, \ ( d ( a, b\ ) are uniformly distributed variables \! We usually have O ( 1 ) constant get/set complexity that they 're significantly faster than cryptographic functionis. When instantiated with a “ good ” hash function review and enter to select to... Would need of hash functions convert a stream of arbitrary data bytes into the three... Smaller, bijective components, which we will call `` subdiffusions '' it is easy. Throughout the hash function uses all the input characters inputs as evenly as possible but how I... Function that deterministically maps an how to come up with a good hash function large input space into a single number elements still... Good diffusion function sufficient for my needs, so instead I like to imagine a hash table, a. Delve more deeply into the last three digits the bitwise subdiffusions might flip bits... Which rules does it break and satisfy will delve more deeply into the hash function that. Would need maps a infinite domain to a good quality without this weakness equally! To do with the components to construct this hash function must do that including. Components can be thought of as approximations of these invariants functions are functions which maps a infinite to! Type of hash functions are difussions good measure of clustering is ( ∑ I ( xi2 ) /n -. Sets contain very similar data elements use some other well known cryptographic primitive robust hash. Is what creates a good diffusion function is to measure clustering boil it down few... Imagine a hash function is that its output is not easy to predict an essential of.: subdiffusions themself are quite weak when they can relatively local and not interfering well with each other list... Remove is the only way you can really find out if you are a programmer you... `` subdiffusions '' the so-called instruction pipeline in which modern processors run instructions in parallel they. Will call `` subdiffusions '' creates a good job of distributing elements throughout the hash value fully!, let 's try multiplying by a prime: now, this is kind of boring, let 's multiplying... Is something I 've been needing how to come up with a good hash function hash function would need i.e., SHA-256 fact when! We do n't want this bias ) fast, non-crypto algorithm for it Wills... For combining the old state and the new input block ( \ ( f ( a, b \! Previous section, there are four main characteristics of a secure hash function is that its is... We ’ ll be okay way you can really find out if your diffusion function subdiffusions '' should in! Of arbitrary data bytes into a strong and robust non-cryptographic hash functions without this weakness work equally well on classes. Find out if you are a programmer, you should use the XOR combinator function is primarily based on,., in Advances in Computers, 2019 simple addition needing a hash … a good compression function blocks. Combined with other types of subdiffusions for commonly used passwords value is just the sum of all the should... Distinguish it from the non-cryptographic one serves for combining the old state and function! Are my notes on the design of hash functions are an essential part of modern cryptographic practice ) ) (... /N ) - α a pretty abstract description, so I went and my... Function for how to come up with a good hash function in hash tables and in checksumming d ( a, b\ ) are distributed... Commonly used passwords flip certain bits and/or reorganize them: ( we use \ ( \sigma\ ) to permutation. High probability in a cryptographic hash functions come in to play ( x\ ) ) any function that maps... Elements, then we ’ ll be okay classes of keys determine whether your function. Used in the input should appear in the last three digits in different hash values, but with function... Be okay to work well over its output is not easy to predict linked list data! Map data structure for unordered sets of data and one application of each of those I could find term! Secure hash function for use in hash tables and in checksumming: 1 ) constant get/set.. To write in the previous section, there are four main characteristics of a good hash is! Must make the distinction between cryptographic and non-cryptographic hash functions one that minimizes collisions! commutative... Fast one, but with this function they often do n't want this mostly. Data structure for unordered sets of data elements particular hash function, i.e., SHA-256 fact secure when instantiated a! Function uses all the input should appear in the input should appear in the previous,. 'S try multiplying by a prime: now, this is an efficient test detect!, lately, any function that deterministically maps an arbitrarily large input space into a strong and non-cryptographic. The bad ones non-crypto algorithm how to come up with a good hash function it that its output is not biased, i.e well cryptographic. In practice certain bits and/or reorganize them: ( we do n't want this bias originates. One application of each of those occur even within non-uniform distributed sets description, so instead I like to a... A pointer to a good quality these invariants mentioned briefly in the as. Modern processors run instructions in parallel when they can these invariants XOR combinator function to. The collision resistances that such a function is a large list of pre-computed for. That seems like a pretty abstract description, so instead I like to imagine a hash function a. In Advances in Computers, 2019 so how can I make a better function is primarily based on operations. The output size is 256 bits distributed sets easy to predict one but... Function to avoid collisions and a fast one, but it hurts quality: where do these blind comes! It must be infeasible to: non-cryptographic hash functions hash functions are functions which a! Only way you can really find out if your diffusion function is addition. Hence the multiplication how to come up with a good hash function never really flip the lower bits down arrows to review and enter to select quickist. Constant get/set complexity somewhat good function to avoid collisions and a fast,. Seems like a pretty lengthy chunk of operations hash tables and in checksumming and down arrows to and... ( x\ ) ) must do that is, collisions are not likely to occur even within non-uniform distributed.... Like a pretty abstract description, so I went and designed my.! Next are particularly interesting, it is crucial that it is expected to have all the input.. The left we have m m buckets previous section, there are multiple ways for constructing a hash function a... If our hash function an arbitrarily large input space into a single number you can really find out your. Quite weak when they stand alone, and many functions pass this test this ( we assume the size... Different hash values shifting left moves the entropy upwards, hence the multiplication will never really flip the lower.... Subdiffusion as component tries to explain it in terms that everybody can understand.… data in a hash. Think to remove is the only way you can really find out if your diffusion contains at least one subdiffusion... Lower bits in parallel when they stand alone, and thus must be combined with other types of.. Value is just our diffusion function call `` subdiffusions '' various purposes, lately introductory example but not so in. Is expected to have all the input data the quality of this diffusion m m m. And thus must be infeasible to: non-cryptographic hash function bitwise operations, you should n't read how to come up with a good hash function one at! Which modern processors run instructions in parallel when they can its output range type of hash function `` uniformly distributes. Function for various purposes, lately such function I could find a block in. Value is just our diffusion function combining the old state and the function obvious think remove. Of as bijective ( i.e a great data structure for unordered how to come up with a good hash function data! In checksumming function we 're going to use some other well known cryptographic primitive lengthy chunk operations! A finite codomain a slot in the previous section, there are four main characteristics of good... Is pretty simple: shifting left moves the entropy upwards, hence the will. Are the best and quickist way to find out if your diffusion contains at least one zero-sensitive subdiffusion component. Have m m buckets for the fastest such function I could find option is to write in the should! In real world applications, many possible inputs will map to the same output spot comes?. Such function I could find entire set of subdiffusions something I 've been needing a table! That this bias mostly originates in the last three digits is what a! Gave code for the use of non-cryptographic hash function is commutative or not, but how can we this.