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I'm working on a system that has 10's of thousands of flags for a user. The flags are all sequential in number, 0 through X, whatever X ends up being. X is expected to grow over time. And we're expecting to have lots and lots of users as well.
Our primary concerns are:
Being able to quickly test whether the user has set any given flag.
Being able to quickly set a flag.
Being able to optimize the data storage to as small a size as possible.
With 10k flags, we're looking at around 1k of data per user, in memory, if we use a bit vector. Which might be too much. And to make matters worse, this is in Javascript, being stored in a document database serialized as JSON, which means that we have several storage options, none of them which I particularly like.
Store flags as the JSON output of the Uint32Array object. Looks like: "{"0":10,"1":4294967295}". Unfortunately needs an average of 17 bytes per 4 bytes stored as the flags approach their filled state, which is over 4x the memory, and leads to about 5k of memory when serialized. This is not ideal.
Perform our own serialization of the JSON, using base64 to avoid the bloated sizes of the numbers-as-strings approach. Unfortunately that adds an extra processing step to the JSON input/output phases which complicates things because now we have to modify our data during the process and will slow everything down.
So... putting aside the bitvector idea for a bit. I was wondering if there's possibly a better approach. I considered using an "array of ranges", something like:
[{"m":0,"x":100},{"m":102},{"m":108,"x":204}]
We can make a few assumptions about the data in this system, which is what led me to this approach:
Flags are never un-set. Once it's set, it will remain set.
Flags are generally clustered. If flag X is set, there's a huge probability that X-1 and X+1 will be set as well.
Flags will generally be set at increasing index values. If Flag X is being set, then X-1 is more likely to be set than X+1, and X+1 is likely to be set fairly soon afterwards.
So because of these conditions, I think storing an array of range objects might be the optimal solution. That way, over time, the user's flags eventually condense down into one large range entry. The optimal case is of course:
[{"m":0,"x":10000}]
The worst case scenario, of course, is if they somehow manage to find themselves in a state where they set every other flag.
[{"m":0},{"m":2},{"m":4},{"m":6},{"m":8},{"m":10}...{"m":10000}]
That would be bad. Far worse than the bitvector solution, I think. But we're pretty confident that won't happen.
So, as to the ability to quickly decide if a flag is set; that's simply an O(logn) binary search (since the array will be sorted); just find the range object closest to your number, check to see if your number is in that range, and return.
Insertions are more tricky. It'll still be a binary search, but now we're modifying the array.
one adjacent sibling insert: optimal scenario. We find a range where the min or max is one off from the number we're inserting, and simply decrement or increment the value on the current range. O(1)
No adjacent siblings insert: simply insert a new node with the min set. O(n), because we'll be moving everything after it in the array downwards.
Two adjacent siblings insert: Change the max to the max value of the right sibling range, delete the right sibling range from the array and shift everything after it to the left. O(n).
So cases 2+3 have me wondering if I shouldn't try to use some sort of balanced binary search tree for this. A Red-Black tree, for example.
Is that worth the bother? Am I overthinking this?
I have a dataset which is a list of prefix ranges, and the prefixes aren't all the same size. Here are a few examples:
low: 54661601 high: 54661679 "bin": a
low: 526219100 high: 526219199 "bin": b
low: 4305870404 high: 4305870404 "bin": c
I want to look up which "bin" corresponds to a particular value with the corresponding prefix. For example, value 5466160179125211 would correspond to "bin" a. In the case of overlaps (of which there are few), we could return either the longest prefix or all prefixes.
The optimal algorithm is clearly some sort of tree into which the bin objects could be inserted, where each successive level of the tree represents more and more of the prefix.
The question is: how do we implement this (in one query) in a database? It is permissible to alter/add to the data set. What would be the best data & query design for this? An answer using mongo or MySQL would be best.
If you make a mild assumption about the number of overlaps in your prefix ranges, it is possible to do what you want optimally using either MongoDB or MySQL. In my answer below, I'll illustrate with MongoDB, but it should be easy enough to port this answer to MySQL.
First, let's rephrase the problem a bit. When you talk about matching a "prefix range", I believe what you're actually talking about is finding the correct range under a lexicographic ordering (intuitively, this is just the natural alphabetic ordering of strings). For instance, the set of numbers whose prefix matches 54661601 to 54661679 is exactly the set of numbers which, when written as strings, are lexicographically greater than or equal to "54661601", but lexicographically less than "54661680". So the first thing you should do is add 1 to all your high bounds, so that you can express your queries this way. In mongo, your documents would look something like
{low: "54661601", high: "54661680", bin: "a"}
{low: "526219100", high: "526219200", bin: "b"}
{low: "4305870404", high: "4305870405", bin: "c"}
Now the problem becomes: given a set of one-dimensional intervals of the form [low, high), how can we quickly find which interval(s) contain a given point? The easiest way to do this is with an index on either the low or high field. Let's use the high field. In the mongo shell:
db.coll.ensureIndex({high : 1})
For now, let's assume that the intervals don't overlap at all. If this is the case, then for a given query point "x", the only possible interval containing "x" is the one with the smallest high value greater than "x". So we can query for that document and check if its low value is also less than "x". For instance, this will print out the matching interval, if there is one:
db.coll.find({high : {'$gt' : "5466160179125211"}}).sort({high : 1}).limit(1).forEach(
function(doc){ if (doc.low <= "5466160179125211") printjson(doc) }
)
Suppose now that instead of assuming the intervals don't overlap at all, you assume that every interval overlaps with less than k neighboring intervals (I don't know what value of k would make this true for you, but hopefully it's a small one). In that case, you can just replace 1 with k in the "limit" above, i.e.
db.coll.find({high : {'$gt' : "5466160179125211"}}).sort({high : 1}).limit(k).forEach(
function(doc){ if (doc.low <= "5466160179125211") printjson(doc) }
)
What's the running time of this algorithm? The indexes are stored using B-trees, so if there are n intervals in your data set, it takes O(log n) time to lookup the first matching document by high value, then O(k) time to iterate over the next k documents, for a total of O(log n + k) time. If k is constant, or in fact anything less than O(log n), then this is asymptotically optimal (this is in the standard model of computation; I'm not counting number of external memory transfers or anything fancy).
The only case where this breaks down is when k is large, for instance if some large interval contains nearly all the other intervals. In this case, the running time is O(n). If your data is structured like this, then you'll probably want to use a different method. One approach is to use mongo's "2d" indexing, with your low and high values codifying x and y coordinates. Then your queries would correspond to querying for points in a given region of the x - y plane. This might do well in practice, although with the current implementation of 2d indexing, the worst case is still O(n).
There are a number of theoretical results that achieve O(log n) performance for all values of k. They go by names such as Priority Search Trees, Segment trees, Interval Trees, etc. However, these are special-purpose data structures that you would have to implement yourself. As far as I know, no popular database currently implements them.
"Optimal" can mean different things to different people. It seems that you could do something like save your low and high values as varchars. Then all you have to do is
select bin from datatable where '5466160179125211' between low and high
Or if you had some reason to keep the values as integers in the table, you could do the CASTing in the query.
I have no idea whether this would give you terrible performance with a large dataset. And I hope I understand what you want to do.
With MySQL you may have to use a stored procedure, which you call to map value to bin. Said procedure would query the list of buckets for each row and do arithmetic or string ops to find the matching bucket. You could improve this design by using fixed length prefixes, arranged in a fixed number of layers. You could assign a fixed depth to your tree and each layer has a table. You won't get tree-like performance with either of these approaches.
If you want to do something more sophisticated, I suspect you have to use a different platform.
Sql Server has a Hierarchy data type:
http://technet.microsoft.com/en-us/library/bb677173.aspx
PostgreSQL has a cidr data type. I'm not familiar with the level of query support it has, but in theory you could build a routing table inside of your db and use that to assign buckets:
http://www.postgresql.org/docs/7.4/static/datatype-net-types.html#DATATYPE-CIDR
Peyton! :)
If you need to keep everything as integers, and want it to work with a single query, this should work:
select bin from datatable where 5466160179125211 between
low*pow(10, floor(log10(5466160179125211))-floor(log10(low)))
and ((high+1)*pow(10, floor(log10(5466160179125211))-floor(log10(high)))-1);
In this case, it would search between the numbers 5466160100000000 (the lowest number with the low prefix & the same number of digits as the number to find) and 546616799999999 (the highest number with the high prefix & the same number of digits as the number to find). This should still work in cases where the high prefix has more digits than the low prefix. It should also work (I think) in cases where the number is shorter than the length of the prefixes, where the varchar code in the previous solution can give incorrect results.
You'll want to experiment to compare the performance of having a lot of inline math in the query (as in this solution) vs. the performance of using varchars.
Edit: Performance seems to be really good either way even on big tables with no indexes; if you can use varchars then you might be able to further boost performance by indexing the low and high columns. Note that you'd definitely want to use varchars if any of the prefixes have initial zeroes. Here's a fix to allow for the case where the number is shorter than the prefix when using varchars:
select * from datatable2 where '5466' between low and high
and length('5466') >= length(high);
I need to keep track of around 10000 elements of an array in my algorithm .So for this I need boolean for each record.If I used char array to keep track of 10000 arrays (as 0/1),it would take up lot of memory.
So Can I create an bit array of 10000 bits in Cuda where each bit represents corresponding array record?
As Roger said, the answer is yes, CUDA provides the same bitwise operations (i.e. >>, << and &) as normal C so you can implement your bit array essentially normally (almost, see thread synchronisation issues below).
However, for your situation it is almost certainly not a good idea.
There are problems with thread syncronisation. Imagine two of the threads on your GPU are inverting two bits of a single entry of your array. Each thread will read the same value out of memory, and apply their operation to it, but the thread that writes its value back to memory last will overwrite the result of the other thread. (Note: if your bit array is not being modified by the GPU code then this isn't a problem.)
And, unless this is explicitly required, you shouldn't be optimising for memory use, an array with 10K elements does not take much memory at all: even if you were storing each boolean in an 64 bit integer it's only 80 KB. And obviously you can store them in a smaller datatype. (You should only start worrying about compressing the array as much as possible when you are getting upwards of tens of millions, or even hundreds of millions of elements.)
Also, the way GPUs work means that you might get best performance by using a reasonably large data type for each boolean (most likely a 32 bit one) so that, for example, memory coalescing works better. (I haven't tested this assertion, you would need to run some benchmarks to check it.)
A long time ago, I bought a data structures book off the bargain table for $1.25. In it, the explanation for a hashing function said that it should ultimately mod by a prime number because of "the nature of math".
What do you expect from a $1.25 book?
Anyway, I've had years to think about the nature of math, and still can't figure it out.
Is the distribution of numbers truly more even when there are a prime number of buckets?
Or is this an old programmer's tale that everyone accepts because everybody else accepts it?
Usually a simple hash function works by taking the "component parts" of the input (characters in the case of a string), and multiplying them by the powers of some constant, and adding them together in some integer type. So for example a typical (although not especially good) hash of a string might be:
(first char) + k * (second char) + k^2 * (third char) + ...
Then if a bunch of strings all having the same first char are fed in, then the results will all be the same modulo k, at least until the integer type overflows.
[As an example, Java's string hashCode is eerily similar to this - it does the characters reverse order, with k=31. So you get striking relationships modulo 31 between strings that end the same way, and striking relationships modulo 2^32 between strings that are the same except near the end. This doesn't seriously mess up hashtable behaviour.]
A hashtable works by taking the modulus of the hash over the number of buckets.
It's important in a hashtable not to produce collisions for likely cases, since collisions reduce the efficiency of the hashtable.
Now, suppose someone puts a whole bunch of values into a hashtable that have some relationship between the items, like all having the same first character. This is a fairly predictable usage pattern, I'd say, so we don't want it to produce too many collisions.
It turns out that "because of the nature of maths", if the constant used in the hash, and the number of buckets, are coprime, then collisions are minimised in some common cases. If they are not coprime, then there are some fairly simple relationships between inputs for which collisions are not minimised. All the hashes come out equal modulo the common factor, which means they'll all fall into the 1/n th of the buckets which have that value modulo the common factor. You get n times as many collisions, where n is the common factor. Since n is at least 2, I'd say it's unacceptable for a fairly simple use case to generate at least twice as many collisions as normal. If some user is going to break our distribution into buckets, we want it to be a freak accident, not some simple predictable usage.
Now, hashtable implementations obviously have no control over the items put into them. They can't prevent them being related. So the thing to do is to ensure that the constant and the bucket counts are coprime. That way you aren't relying on the "last" component alone to determine the modulus of the bucket with respect to some small common factor. As far as I know they don't have to be prime to achieve this, just coprime.
But if the hash function and the hashtable are written independently, then the hashtable doesn't know how the hash function works. It might be using a constant with small factors. If you're lucky it might work completely differently and be nonlinear. If the hash is good enough, then any bucket count is just fine. But a paranoid hashtable can't assume a good hash function, so should use a prime number of buckets. Similarly a paranoid hash function should use a largeish prime constant, to reduce the chance that someone uses a number of buckets which happens to have a common factor with the constant.
In practice, I think it's fairly normal to use a power of 2 as the number of buckets. This is convenient and saves having to search around or pre-select a prime number of the right magnitude. So you rely on the hash function not to use even multipliers, which is generally a safe assumption. But you can still get occasional bad hashing behaviours based on hash functions like the one above, and prime bucket count could help further.
Putting about the principle that "everything has to be prime" is as far as I know a sufficient but not a necessary condition for good distribution over hashtables. It allows everybody to interoperate without needing to assume that the others have followed the same rule.
[Edit: there's another, more specialized reason to use a prime number of buckets, which is if you handle collisions with linear probing. Then you calculate a stride from the hashcode, and if that stride comes out to be a factor of the bucket count then you can only do (bucket_count / stride) probes before you're back where you started. The case you most want to avoid is stride = 0, of course, which must be special-cased, but to avoid also special-casing bucket_count / stride equal to a small integer, you can just make the bucket_count prime and not care what the stride is provided it isn't 0.]
The first thing you do when inserting/retreiving from hash table is to calculate the hashCode for the given key and then find the correct bucket by trimming the hashCode to the size of the hashTable by doing hashCode % table_length. Here are 2 'statements' that you most probably have read somewhere
If you use a power of 2 for table_length, finding (hashCode(key) % 2^n ) is as simple and quick as (hashCode(key) & (2^n -1)). But if your function to calculate hashCode for a given key isn't good, you will definitely suffer from clustering of many keys in a few hash buckets.
But if you use prime numbers for table_length, hashCodes calculated could map into the different hash buckets even if you have a slightly stupid hashCode function.
And here is the proof.
If suppose your hashCode function results in the following hashCodes among others {x , 2x, 3x, 4x, 5x, 6x...}, then all these are going to be clustered in just m number of buckets, where m = table_length/GreatestCommonFactor(table_length, x). (It is trivial to verify/derive this). Now you can do one of the following to avoid clustering
Make sure that you don't generate too many hashCodes that are multiples of another hashCode like in {x, 2x, 3x, 4x, 5x, 6x...}.But this may be kind of difficult if your hashTable is supposed to have millions of entries.
Or simply make m equal to the table_length by making GreatestCommonFactor(table_length, x) equal to 1, i.e by making table_length coprime with x. And if x can be just about any number then make sure that table_length is a prime number.
From - http://srinvis.blogspot.com/2006/07/hash-table-lengths-and-prime-numbers.html
http://computinglife.wordpress.com/2008/11/20/why-do-hash-functions-use-prime-numbers/
Pretty clear explanation, with pictures too.
Edit: As a summary, primes are used because you have the best chance of obtaining a unique value when multiplying values by the prime number chosen and adding them all up. For example given a string, multiplying each letter value with the prime number and then adding those all up will give you its hash value.
A better question would be, why exactly the number 31?
Just to put down some thoughts gathered from the answers.
Hashing uses modulus so any value can fit into a given range
We want to randomize collisions
Randomize collision meaning there are no patterns as how collisions would happen, or, changing a small part in input would result a completely different hash value
To randomize collision, avoid using the base (10 in decimal, 16 in hex) as modulus, because 11 % 10 -> 1, 21 % 10 -> 1, 31 % 10 -> 1, it shows a clear pattern of hash value distribution: value with same last digits will collide
Avoid using powers of base (10^2, 10^3, 10^n) as modulus because it also creates a pattern: value with same last n digits matters will collide
Actually, avoid using any thing that has factors other than itself and 1, because it creates a pattern: multiples of a factor will be hashed into selected values
For example, 9 has 3 as factor, thus 3, 6, 9, ...999213 will always be hashed into 0, 3, 6
12 has 3 and 2 as factor, thus 2n will always be hashed into 0, 2, 4, 6, 8, 10, and 3n will always be hashed into 0, 3, 6, 9
This will be a problem if input is not evenly distributed, e.g. if many values are of 3n, then we only get 1/3 of all possible hash values and collision is high
So by using a prime as a modulus, the only pattern is that multiple of the modulus will always hash into 0, otherwise hash values distributions are evenly spread
tl;dr
index[hash(input)%2] would result in a collision for half of all possible hashes and a range of values. index[hash(input)%prime] results in a collision of <2 of all possible hashes. Fixing the divisor to the table size also ensures that the number cannot be greater than the table.
Primes are used because you have good chances of obtaining a unique value for a typical hash-function which uses polynomials modulo P.
Say, you use such hash-function for strings of length <= N, and you have a collision. That means that 2 different polynomials produce the same value modulo P. The difference of those polynomials is again a polynomial of the same degree N (or less). It has no more than N roots (this is here the nature of math shows itself, since this claim is only true for a polynomial over a field => prime number). So if N is much less than P, you are likely not to have a collision. After that, experiment can probably show that 37 is big enough to avoid collisions for a hash-table of strings which have length 5-10, and is small enough to use for calculations.
Just to provide an alternate viewpoint there's this site:
http://www.codexon.com/posts/hash-functions-the-modulo-prime-myth
Which contends that you should use the largest number of buckets possible as opposed to to rounding down to a prime number of buckets. It seems like a reasonable possibility. Intuitively, I can certainly see how a larger number of buckets would be better, but I'm unable to make a mathematical argument of this.
It depends on the choice of hash function.
Many hash functions combine the various elements in the data by multiplying them with some factors modulo the power of two corresponding to the word size of the machine (that modulus is free by just letting the calculation overflow).
You don't want any common factor between a multiplier for a data element and the size of the hash table, because then it could happen that varying the data element doesn't spread the data over the whole table. If you choose a prime for the size of the table such a common factor is highly unlikely.
On the other hand, those factors are usually made up from odd primes, so you should also be safe using powers of two for your hash table (e.g. Eclipse uses 31 when it generates the Java hashCode() method).
Copying from my other answer https://stackoverflow.com/a/43126969/917428. See it for more details and examples.
I believe that it just has to do with the fact that computers work with in base 2. Just think at how the same thing works for base 10:
8 % 10 = 8
18 % 10 = 8
87865378 % 10 = 8
It doesn't matter what the number is: as long as it ends with 8, its modulo 10 will be 8.
Picking a big enough, non-power-of-two number will make sure the hash function really is a function of all the input bits, rather than a subset of them.
"The nature of math" regarding prime power moduli is that they are one building block of a finite field. The other two building blocks are an addition and a multiplication operation. The special property of prime moduli is that they form a finite field with the "regular" addition and multiplication operations, just taken to the modulus. This means every multiplication maps to a different integer modulo the prime, so does every addition.
Prime moduli are advantageous because:
They give the most freedom when choosing the secondary multiplier in secondary hashing, all multipliers except 0 will end up visiting all elements exactly once
If all hashes are less than the modulus there will be no collisions at all
Random primes mix better than power of two moduli and compress the information of all the bits not just a subset
They however have a big downside, they require an integer division, which takes many (~ 15-40) cycles, even on a modern CPU. With around half the computation one can make sure the hash is mixed up very well. Two multiplications and xorshift operations will mix better than a prime moudulus. Then we can use whatever hash table size and hash reduction is fastest, giving 7 operations in total for power of 2 table sizes and around 9 operations for arbitrary sizes.
I recently looked at many of the fastest hash table implementations and most of them don't use prime moduli.
The distribution of the hash table indices are mainly dependent on the hash function in use. A prime modulus can't fix a bad hash function and a good hash function does not benefit from a prime modulus. There are cases where they can be advantageous however. It can mend a half-bad hash function for example.
Primes are unique numbers. They are
unique in that, the product of a prime
with any other number has the best
chance of being unique (not as unique
as the prime itself of-course) due to
the fact that a prime is used to
compose it. This property is used in
hashing functions.
Given a string “Samuel”, you can
generate a unique hash by multiply
each of the constituent digits or
letters with a prime number and adding
them up. This is why primes are used.
However using primes is an old
technique. The key here to understand
that as long as you can generate a
sufficiently unique key you can move
to other hashing techniques too. Go
here for more on this topic about
http://www.azillionmonkeys.com/qed/hash.html
http://computinglife.wordpress.com/2008/11/20/why-do-hash-functions-use-prime-numbers/
Suppose your table-size (or the number for modulo) is T = (B*C). Now if hash for your input is like (N*A*B) where N can be any integer, then your output won't be well distributed. Because every time n becomes C, 2C, 3C etc., your output will start repeating. i.e. your output will be distributed only in C positions. Note that C here is (T / HCF(table-size, hash)).
This problem can be eliminated by making HCF 1. Prime numbers are very good for that.
Another interesting thing is when T is 2^N. These will give output exactly same as all the lower N bits of input-hash. As every number can be represented powers of 2, when we will take modulo of any number with T, we will subtract all powers of 2 form number, which are >= N, hence always giving off number of specific pattern, dependent on the input. This is also a bad choice.
Similarly, T as 10^N is bad as well because of similar reasons (pattern in decimal notation of numbers instead of binary).
So, prime numbers tend to give a better distributed results, hence are good choice for table size.
I would say the first answer at this link is the clearest answer I found regarding this question.
Consider the set of keys K = {0,1,...,100} and a hash table where the number of buckets is m = 12. Since 3 is a factor of 12, the keys that are multiples of 3 will be hashed to buckets that are multiples of 3:
Keys {0,12,24,36,...} will be hashed to bucket 0.
Keys {3,15,27,39,...} will be hashed to bucket 3.
Keys {6,18,30,42,...} will be hashed to bucket 6.
Keys {9,21,33,45,...} will be hashed to bucket 9.
If K is uniformly distributed (i.e., every key in K is equally likely to occur), then the choice of m is not so critical. But, what happens if K is not uniformly distributed? Imagine that the keys that are most likely to occur are the multiples of 3. In this case, all of the buckets that are not multiples of 3 will be empty with high probability (which is really bad in terms of hash table performance).
This situation is more common that it may seem. Imagine, for instance, that you are keeping track of objects based on where they are stored in memory. If your computer's word size is four bytes, then you will be hashing keys that are multiples of 4. Needless to say that choosing m to be a multiple of 4 would be a terrible choice: you would have 3m/4 buckets completely empty, and all of your keys colliding in the remaining m/4 buckets.
In general:
Every key in K that shares a common factor with the number of buckets m will be hashed to a bucket that is a multiple of this factor.
Therefore, to minimize collisions, it is important to reduce the number of common factors between m and the elements of K. How can this be achieved? By choosing m to be a number that has very few factors: a prime number.
FROM THE ANSWER BY Mario.
I'd like to add something for Steve Jessop's answer(I can't comment on it since I don't have enough reputation). But I found some helpful material. His answer is very help but he made a mistake: the bucket size should not be a power of 2. I'll just quote from the book "Introduction to Algorithm" by Thomas Cormen, Charles Leisersen, et al on page263:
When using the division method, we usually avoid certain values of m. For example, m should not be a power of 2, since if m = 2^p, then h(k) is just the p lowest-order bits of k. Unless we know that all low-order p-bit patterns are equally likely, we are better off designing the hash function to depend on all the bits of the key. As Exercise 11.3-3 asks you to show, choosing m = 2^p-1 when k is a character string interpreted in radix 2^p may be a poor choice, because permuting the characters of k does not change its hash value.
Hope it helps.
This question was merged with the more appropriate question, why hash tables should use prime sized arrays, and not power of 2.
For hash functions itself there are plenty of good answers here, but for the related question, why some security-critical hash tables, like glibc, use prime-sized arrays, there's none yet.
Generally power of 2 tables are much faster. There the expensive h % n => h & bitmask, where the bitmask can be calculated via clz ("count leading zeros") of the size n. A modulo function needs to do integer division which is about 50x slower than a logical and. There are some tricks to avoid a modulo, like using Lemire's https://lemire.me/blog/2016/06/27/a-fast-alternative-to-the-modulo-reduction/, but generally fast hash tables use power of 2, and secure hash tables use primes.
Why so?
Security in this case is defined by attacks on the collision resolution strategy, which is with most hash tables just linear search in a linked list of collisions. Or with the faster open-addressing tables linear search in the table directly. So with power of 2 tables and some internal knowledge of the table, e.g. the size or the order of the list of keys provided by some JSON interface, you get the number of right bits used. The number of ones on the bitmask. This is typically lower than 10 bits. And for 5-10 bits it's trivial to brute force collisions even with the strongest and slowest hash functions. You don't get the full security of your 32bit or 64 bit hash functions anymore. And the point is to use fast small hash functions, not monsters such as murmur or even siphash.
So if you provide an external interface to your hash table, like a DNS resolver, a programming language, ... you want to care about abuse folks who like to DOS such services. It's normally easier for such folks to shut down your public service with much easier methods, but it did happen. So people did care.
So the best options to prevent from such collision attacks is either
1) to use prime tables, because then
all 32 or 64 bits are relevant to find the bucket, not just a few.
the hash table resize function is more natural than just double. The best growth function is the fibonacci sequence and primes come closer to that than doubling.
2) use better measures against the actual attack, together with fast power of 2 sizes.
count the collisions and abort or sleep on detected attacks, which is collision numbers with a probability of <1%. Like 100 with 32bit hash tables. This is what e.g. djb's dns resolver does.
convert the linked list of collisions to tree's with O(log n) search not O(n) when an collision attack is detected. This is what e.g. java does.
There's a wide-spread myth that more secure hash functions help to prevent such attacks, which is wrong as I explained. There's no security with low bits only. This would only work with prime-sized tables, but this would use a combination of the two slowest methods, slow hash plus slow prime modulo.
Hash functions for hash tables primarily need to be small (to be inlinable) and fast. Security can come only from preventing linear search in the collisions. And not to use trivially bad hash functions, like ones insensitive to some values (like \0 when using multiplication).
Using random seeds is also a good option, people started with that first, but with enough information of the table even a random seed does not help much, and dynamic languages typically make it trivial to get the seed via other methods, as it's stored in known memory locations.
For a hash function it's not only important to minimize colisions generally but to make it impossible to stay with the same hash while chaning a few bytes.
Say you have an equation:
(x + y*z) % key = x with 0<x<key and 0<z<key.
If key is a primenumber n*y=key is true for every n in N and false for every other number.
An example where key isn't a prime example:
x=1, z=2 and key=8
Because key/z=4 is still a natural number, 4 becomes a solution for our equation and in this case (n/2)*y = key is true for every n in N. The amount of solutions for the equation have practially doubled because 8 isn't a prime.
If our attacker already knows that 8 is possible solution for the equation he can change the file from producing 8 to 4 and still gets the same hash.
I've read the popular wordpress website linked in some of the above popular answers at the top. From what I've understood, I'd like to share a simple observation I made.
You can find all the details in the article here, but assume the following holds true:
Using a prime number gives us the "best chance" of an unique value
A general hashmap implementation wants 2 things to be unique.
Unique hash code for the key
Unique index to store the actual value
How do we get the unique index? By making the initial size of the internal container a prime as well. So basically, prime is involved because it possesses this unique trait of producing unique numbers which we end up using to ID objects and finding indexes inside the internal container.
Example:
key = "key"
value = "value"
uniqueId = "k" * 31 ^ 2 +
"e" * 31 ^ 1` +
"y"
maps to unique id
Now we want a unique location for our value - so we
uniqueId % internalContainerSize == uniqueLocationForValue , assuming internalContainerSize is also a prime.
I know this is simplified, but I'm hoping to get the general idea through.
I'm using sequential ids as primary keys and there are cases where I don't want those ids to be visible to users, for example I might want to avoid urls like ?invoice_id=1234 that allow users to guess how many invoices the system as a whole is issuing.
I could add a database field with a GUID or something conjured up from hash functions, random strings and/or numeric base conversions, but schemes of that kind have three issues that I find annoying:
Having to allocate the extra database field. I know I could use the GUID as my primary key, but my auto-increment integer PK's are the right thing for most purposes, and I don't want to change that.
Having to think about the possibility of hash/GUID collisions. I give my full assent to all the arguments about GUID collisions being as likely as spontaneous combustion or whatever, but disregarding exceptional cases because they're exceptional goes against everything else I've been taught, and it continues to bother me even when I know I should be more bothered about other things.
I don't know how to safely trim hash-based identifiers, so even if my private ids are 16 or 32 bits, I'm stuck with 128 bit generated identifiers that are a nuisance in urls.
I'm interested in 1-1 mappings of an id range, stretchable or shrinkable so that for example 16-bit ids are mapped to 16 bit ids, 32 bit ids mapped to 32 bit ids, etc, and that would stop somebody from trying to guess the total number of ids allocated or the rate of id allocation over a period.
For example, if my user ids are 16 bit integers (0..65535), then an example of a transformation that somewhat obfuscates the id allocation is the function f(x) = (x mult 1001) mod 65536. The internal id sequence of 1, 2, 3 becomes the public id sequence of 1001, 2002, 3003. With a further layer of obfuscation from base conversion, for example to base 36, the sequence becomes 'rt', '1jm', '2bf'. When the system gets a request to the url ?userid=2bf, it converts from base 36 to get 3003 and it applies the inverse transformation g(x) = (x mult 1113) mod 65536 to get back to the internal id=3.
A scheme of that kind is enough to stop casual observation by casual users, but it's easily solvable by someone who's interested enough to try to puzzle it through. Can anyone suggest something that's a bit stronger, but is easily implementable in say PHP without special libraries? This is getting close to a roll-your-own encryption scheme, so maybe there is a proper encryption algorithm that's widely available and has the stretchability property mentioned above?
EDIT: Stepping back a little bit, some discussion at codinghorror about choosing from three kinds of keys - surrogate (guid-based), surrogate (integer-based), natural. In those terms, I'm trying to hide an integer surrogate key from users but I'm looking for something shrinkable that makes urls that aren't too long, which I don't know how to do with the standard 128-bit GUID. Sometimes, as commenter Princess suggests below, the issue can be sidestepped with a natural key.
EDIT 2/SUMMARY:
Given the constraints of the question I asked (stretchability, reversibility, ease of implementation), the most suitable solution so far seems to be the XOR-based obfuscation suggested by Someone and Breton.
It would be irresponsible of me to assume that I can achieve anything more than obfuscation/security by obscurity. The knowledge that it's an integer sequence is probably a crib that any competent attacker would be able to take advantage of.
I've given some more thought to the idea of the extra database field. One advantage of the extra field is that it makes it a lot more straightforward for future programmers who are trying to familiarise themselves with the system by looking at the database. Otherwise they'd have to dig through the source code (or documentation, ahem) to work out how a request to a given url is resolved to a given record in the database.
If I allow the extra database field, then some of the other assumptions in the question become irrelevant (for example the transformation doesn't need to be reversible). That becomes a different question, so I'll leave it there.
I find that simple XOR encryption is best suited for URL obfuscation. You can continue using whatever serial number you are using without change. Further XOR encryption doesn't increase the length of source string. If your text is 22 bytes, the encrypted string will be 22 bytes too. It's not easy enough as to be guessed like rot 13 but not heavy weight like DSE/RSA.
Search the net for PHP XOR encryption to find some implementation. The first one I found is here.
I've toyed with this sort of thing myself, in my amateurish way, and arrived at a kind of kooky number scrambling algorithm, involving mixed radices. Basically I have a function that maps a number between 0-N to another number in the 0-N range. For URLS I then map that number to a couple of english words. (words are easier to remember).
A simplified version of what I do, without mixed radices: You have a number that is 32 bits, so ahead of time, have a passkey which is 32-bits long, and XOR the passkey with your input number. Then shuffle the bits around in a determinate reordering. (possibly based on your passkey).
The nice thing about this is
No collisions, as long as you shuffle and xor the same way each time
No need to store the obfuscated keys in the database
Still use your ordered IDS internally, since you can reverse the obfuscation
You can repeat the operation several times to get more obfuscated results.
if you're up for the mixed radix version, it's basically the same, except that I add the steps of converting the input to a mixed raddix number, using the maximum range's prime factors as the digit's bases. Then I shuffle the digits around, keeping the bases with the digits, and turn it back into a standard integer.
You might find it useful to revisit the idea of using a GUID, because you can construct GUIDs in a way that isn't subject to collision.
Check out the Wikipedia page on GUIDs - the "Type 1" algorithm uses both the MAC address of the PC, and the current date/time as inputs. This guarantees that collisions are simply impossible.
Alternatively, if you create a GUID column in your database as an alternative-key (keep using your auto-increment primary keys), define it as unique. Then, if your GUID generation approach does give a duplicate, you'll get an appropriate error on insert that you can handle.
I saw this question yesterday: how reddit generates an alphanum id
I think it's a reasonably good method (and particularily clever)
it uses Python
def to_base(q, alphabet):
if q < 0: raise ValueError, "must supply a positive integer"
l = len(alphabet)
converted = []
while q != 0:
q, r = divmod(q, l)
converted.insert(0, alphabet[r])
return "".join(converted) or '0'
def to36(q):
return to_base(q, '0123456789abcdefghijklmnopqrstuvwxyz')
Add a char(10) field to your order table... call it 'order_number'.
After you create a new order, randomly generate an integer from 1...9999999999. Check to see if it exists in the database under 'order_number'. If not, update your latest row with this value. If it does exist, pick another number at random.
Use 'order_number' for publicly viewable URLs, maybe always padded with zeros.
There's a race condition concern for when two threads attempt to add the same number at the same time... you could do a table lock if you were really concerned, but that's a big hammer. Add a second check after updating, re-select to ensure it's unique. Call recursively until you get a unique entry. Dwell for a random number of milliseconds between calls, and use the current time as a seed for the random number generator.
Swiped from here.
UPDATED As with using the GUID aproach described by Bevan, if the column is constrained as unique, then you don't have to sweat it. I guess this is no different that using a GUID, except that the customer and Customer Service will have an easier time referring to the order.
I've found a much simpler way. Say you want to map N digits, pseudorandomly to N digits. you find the next highest prime from N, and you make your function
prandmap(x) return x * nextPrime(N) % N
this will produce a function that repeats (or has a period) every N, no number is produced twice until x=N+1. It always starts at 0, but is pseudorandom thereafter.
I honestly thing encrypting/decrypting query string data is a bad approach to this problem. The easiest solution is sending data using POST instead of GET. If users are clicking on links with querystring data, you have to resort to some javascript hacks to send data by POST (keep accessibility in mind for users with Javascript turned off). This doesn't prevent users from viewing source, but at the very least it keeps sensitive from being indexed by search engines, assuming the data you're trying to hide really that sensitive in the first place.
Another approach is to use a natural unique key. For example, if you're issuing invoices to customers on a monthly basis, then "yyyyMM[customerID]" uniquely identifies a particular invoice for a particular user.
From your description, personally, I would start off by working with whatever standard encryption library is available (I'm a Java programmer, but I assume, say, a basic AES encryption library must be available for PHP):
on the database, just key things as you normally would
whenever you need to transmit a key to/from a client, use a fairly strong, standard encryption system (e.g. AES) to convert the key to/from a string of garbage. As your plain text, use a (say) 128-byte buffer containing: a (say) 4-byte key, 60 random bytes, and then a 64-byte medium-quality hash of the previous 64 bytes (see Numerical Recipes for an example)-- obviously when you receive such a string, you decrypt it then check if the hash matches before hitting the DB. If you're being a bit more paranoid, send an AES-encrypted buffer of random bytes with your key in an arbitrary position, plus a secure hash of that buffer as a separate parameter. The first option is probably a reasonable tradeoff between performance and security for your purposes, though, especially when combined with other security measures.
the day that you're processing so many invoices a second that AES encrypting them in transit is too performance expensive, go out and buy yourself a big fat server with lots of CPUs to celebrate.
Also, if you want to hide that the variable is an invoice ID, you might consider calling it something other than "invoice_id".