I'm looking for an algorithm that can do a one-to-one mapping of a string onto another string.
I want an algorithm that given an alphabet I can perform a symmetric mapping function.
For example:
Let's consider that I have the alphabet "A","B","C","D","E","F". I want something like F("ABC") = "CEA" and F("CEA") = "ABC" for every N letter permutation.
Surely, an algorithm like this exists. If you know of an algorithm, please post the name of it and I can research it. If I haven't been clear enough in my request, please let me know.
Thanks in advance.
Edit 1:
I should clarify that I want enough entropy so that F("ABC") would equal "CEA" and F("CEA") = "ABC" but then I do NOT want F("ABD") to equal "CEF". Notice how two input letters stayed the same and the two corresponding output letters stayed the same?
So a Caesar Cipher/ROT13 or shuffling the array would not be sufficient. However, I don't need any "real" security. Just enough entropy for the output of the function to appear random. Weak encryption algorithms welcome.
Just create an array of objects that contain 2 fields -- a letter, and a random number. Sort the array. By the random numbers. This creates a mapping where the i-th letter of the alphabet now maps to the i-th letter in the array.
If simple transposition or substitution isn't quite enough, it sounds like you want to advance to a polyalphabetic cipher. The Vigenère cipher is extremely easy to implement in code, but is still difficult to break without using a computer.
I suggest the following.
Perform a dense coding of the input to positive integers - with an alphabet size of n and string length of m you can code the string into integers between zero and n^m - 1. In your example this would be the range [0,215]. Now perform a fixed involution on the encoded number and decode it again.
Take RC4, settle for some password, and you're done. (Not that this would be very safe.)
Take the set of all permutations of your alphabet, shuffle it, and map the first half of the set onto the second half. Bad for large alphabets, of course. :)
Nah, thought that over, I forgot about character repetitions. Maybe divide the input into chunks without repeating chars and apply my suggestion to all of those chunks.
I would restate your problem thus, and give you a strategy for that restatement:
"A substitution cypher where a change in input leads to a larger change in output".
The blocking of characters is irrelevant-- in the end, it's just mappings between numbers. I'll speak of letters here, but you can extend it to any block of n characters.
One of the easiest routes for this is a rotating substitution based on input. Since you already looked at the Vigenere cipher, it should be easy to understand. Instead of making the key be static, have it be dependent on the previous letter. That is, rotate through substitutions a different amount per each input.
The variable rotation satisfies the condition of making each small change push out to a larger change. Note that the algorithm will only push changes in one direction such that changes towards the end have smaller effects. You could run the algorithm both ways (front-to-back, then back-to-front) so that every letter of cleartext changed has the possibility of changing the entire string.
The internal rotation strategy elides the need for keys, while of course losing of most of the cryptographic security. It makes sense in context, though, as you are aiming for entropy rather than security.
You can solve this problem with Format-preserving encryption.
One Java-Library can be found under https://github.com/EVGStudents/FPE.git. There you can define a Regex and encrypt/decrypt string values matching this regex.
Related
DISCLAIMER: I am not asking how to make a URL shortener (I have already implemented the "bijective function" answer found HERE that uses a base-62 encoded string). Instead, I want to expand this implementation to obfuscate the generated string so that it is both:
A) not an easily guessable sequence, and
B) still bijective.
You can easily randomize your base-62 character set, but the problem is that it still increments like any other number in any other base. For example, one possible incremental progression might be {aX9fgE, aX9fg3, aX9fgf, aX9fgR, … ,}
I have come up with an obfuscation technique that I am pleased with in terms of requirement A), but I'm only partially sure that it satisfies B). The idea is this:
The only thing that is guaranteed to change in the incremental approach is the "1's place" (I'll use decimal terminology for practicality reasons). In the sample progression I gave earlier, that would be {E, 3, f, R, …}. So if each character in the base-62 set had its own unique offset number (say, its distance from the "zero character"), then you could apply the offset of the "1's place" character to the rest of the string.
For instance, let's assume a base-5 set with characters {A, f, 9, p, Z, 3} (in ascending order from 0 to 5). Each one would then have a unique offset of 0 to 5 respectively. Counting would look like {A, f, 9, p, Z, 3, fA, ff, f9, fp, …} and so on. So the algorithm, when given a value of fZ3p, would look at the p and, having an offset of +3, would permute the string into Zf9p (assuming the base-5 set is a circular array). The next incremental number would be fZ3Z, and with Z's offset being +4, the algorithm returns 39pZ. These permutated results would be handed off to the user as his/her "unique URL", who would never see the actual base-62 encoded string.
This approach certainly seems reversible; just look at the last character, and perform the same permutation with the negative offset. And I'm thinking that for this reason, it has to still be bijective. But I don't know if this is necessarily true? Are there any edge/corner cases I'm not considering?
EDIT : My intentions are more heavily weighed towards the length of the shortened-URL rather than the security of the pattern. I realize there are plenty of solutions involving cryptographic functions, block ciphers, etc. But I would like to emphasize that I am not asking the best way to achieve A), but rather, "is my offset-approach satisfying B)".
Any holes you can find would be appreciated.
If you honestly want them to be hard to guess, keep it simple.
Start with a normal encryption algorithm running in counter mode. When you get a URL to shorten, increment your counter, encrypt it, convert the result to something using printable characters (e.g., base 64) and put the original URL and the shortened version into your table so you can get the original URL from the shortened version when needed.
The only real question at that point is what encryption algorithm to use. That, in turn, depends on your threat model. I don't see exactly what you gain by making shortened URLs hard to guess, so I'm a bit uncertain about the threat model.
If you want to make it mildly difficult to guess, you could use something like a 40-bit version of RC4. This is pretty easy to break, but enough to keep most people from bothering.
If you want a bit more security, you could step up to DES. That's been broken, but even at this late date breaking it is quite a bit of work.
If you want more security than that, you can use AES.
Note that as you increase the security, the shortened URL gets longer though. RC4-40 starts out with 5 bytes, DES 7 bytes, and AES with 32 bytes. Depending on how you convert to printable text, that's going to expand at least a little.
Another option is to use the Luby-Rackoff construction (see also here), which is a way to generate a pseudo-random permutation from a pseudo-random function.
You just have to pick a "round function" F. F must take as input a key K and a block of bits half the size of what you are encoding. F must produce as output a block of bits also half the size of whatever you are encoding.
Then you just run the Luby-Rackoff construction (aka. "Feistel network") for four rounds, each round using a different K.
The construction guarantees that the result is a bijective map, and it will be hard to invert provided that F is hard to invert.
I tried to solve the same problem (in php) and ended up with those functions:
hashing the row id with a kind of feistel algo
applying a bijective function to compress the integer
So for the A): it's not easily guessable (to me) as you cant increment a string to get the next record without an algo
And for the B): for what I understand it's 100% bijective.
Thanks to #Nemo for naming the feistel network, which lead me to the first function i have linked to.
If you're trying to avoid people crawling the URLs, I think Nick Johnson has the right idea, that you need to make sure your URL space is not dense.
Here's a simple idea: take your URL, and prepend a few random characters to it. Then run it through a compression algorithm -- I'd try range encoding (you can probably specify the basis if you find a good library). This should be decompressible to the original form, and should both impact locality and make the encoded space more sparse.
That said, I imagine nearly all URL shorteners out there keep a hash table with state on the server side. How else are you going to losslessly compress a hundred-character URL into 5 or 6 characters?
I'm looking at ways to deterministically replace unique strings with unique and optimally short replacements. So I have a finite set of strings, and the best compression I could achieve so far is through an enumeration algorithm, where I order the input set and then replace the strings with an enumeration of char strings over an extended alphabet (a..z, A...Z, aa...zz, aA... zZ, a0...z9, Aa..., aaa...zaa, aaA...zaaA, ....).
This works wonderfully as far as compression is concerned, but has the severe drawback that it is not atomic on any given input string. Rather, its result depends on knowing all input strings right from the start, and on the ordering of the input set.
Anybody knows of an algorithm that has similar compression but doesn't require knowing all input strings upfront?! Hashing for example would not work for me, as depending on the size of the input set I'd need a hash length of 8-12 for the hashes to be unique, and that would be too long as replacements (currently, the replacement strings are 1-3 chars long for my use cases (<10,000 input strings)). Also, if theoreticians among us know this is wasted effort, I would be interested to hear :-) .
You could use your enumeration scheme, but sorted by the order in which you first encounter the input strings.
For example, the first string you ever process can be mapped to "a".
The next distinct string would be mapped to "b", etc.
Every time you process a string, you'd need to look it up to see if it has already been mapped.
"Optimally short" depends on the population of strings from which your samples are drawn. In the absence of systematic redundancy in the population, you will find that only a fraction of arbitrary strings can be compressed at all (e.g., consider trying to compress random bit strings).
If you can make assumptions about your data, such as "the strings are expected to be mainly composed of English words" then you can do something simple and effective based on letter frequency (e.g., for English, the relative frequency order is something like ETAOINSHRDLUGCY..., so you would want to use fewer bits to represent Es and more bits to represent uncommon letters like Q).
Cheers.
I would like an algorithm for a function that takes n integers and returns one integer. For small changes in the input, the resulting integer should vary greatly. Even though I've taken a number of courses in math, I have not used that knowledge very much and now I need some help...
An important property of this function should be that if it is used with coordinate pairs as input and the result is plotted (as a grayscale value for example) on an image, any repeating patterns should only be visible if the image is very big.
I have experimented with various algorithms for pseudo-random numbers with little success and finally it struck me that md5 almost meets my criteria, except that it is not for numbers (at least not from what I know). That resulted in something like this Python prototype (for n = 2, it could easily be changed to take a list of integers of course):
import hashlib
def uniqnum(x, y):
return int(hashlib.md5(str(x) + ',' + str(y)).hexdigest()[-6:], 16)
But obviously it feels wrong to go over strings when both input and output are integers. What would be a good replacement for this implementation (in pseudo-code, python, or whatever language)?
A "hash" is the solution created to solve exactly the problem you are describing. See wikipedia's article
Any hash function you use will be nice; hash functions tend to be judged based on these criteria:
The degree to which they prevent collisions (two separate inputs producing the same output) -- a by-product of this is the degree to which the function minimizes outputs that may never be reached from any input.
The uniformity the distribution of its outputs given a uniformly distributed set of inputs
The degree to which small changes in the input create large changes in the output.
(see perfect hash function)
Given how hard it is to create a hash function that maximizes all of these criteria, why not just use one of the most commonly used and relied-on existing hash functions there already are?
From what it seems, turning integers into strings almost seems like another layer of encryption! (which is good for your purposes, I'd assume)
However, your question asks for hash functions that deal specifically with numbers, so here we go.
Hash functions that work over the integers
If you want to borrow already-existing algorithms, you may want to dabble in pseudo-random number generators
One simple one is the middle square method:
Take a digit number
Square it
Chop off the digits and leave the middle digits with the same length as your original.
ie,
1111 => 01234321 => 2342
so, 1111 would be "hashed" to 2342, in the middle square method.
This way isn't that effective, but for a few number of hashes, this has very low collision rates, a uniform distribution, and great chaos-potential (small changes => big changes). But if you have many values, time to look for something else...
The grand-daddy of all feasibly efficient and simple random number generators is the (Mersenne Twister)[http://en.wikipedia.org/wiki/Mersenne_twister]. In fact, an implementation is probably out there for every programming language imaginable. Your hash "input" is something that will be called a "seed" in their terminology.
In conclusion
Nothing wrong with string-based hash functions
If you want to stick with the integers and be fancy, try using your number as a seed for a pseudo-random number generator.
Hashing fits your requirements perfectly. If you really don't want to use strings, find a Hash library that will take numbers or binary data. But using strings here looks OK to me.
Bob Jenkins' mix function is a classic choice, at when n=3.
As others point out, hash functions do exactly what you want. Hashes take bytes - not character strings - and return bytes, and converting between integers and bytes is, of course, simple. Here's an example python function that works on 32 bit integers, and outputs a 32 bit integer:
import hashlib
import struct
def intsha1(ints):
input = struct.pack('>%di' % len(ints), *ints)
output = hashlib.sha1(input).digest()
return struct.unpack('>i', output[:4])
It can, of course, be easily adapted to work with different length inputs and outputs.
Have a look at this, may be you can be inspired
Chaotic system
In chaotic dynamics, small changes vary results greatly.
A x-bit block cipher will take an number and convert it effectively to another number. You could combine (sum/mult?) your input numbers and cipher them, or iteratively encipher each number - similar to a CBC or chained mode. Google 'format preserving encyption'. It is possible to create a 32-bit block cipher (not widely 'available') and use this to create a 'hashed' output. Main difference between hash and encryption, is that hash is irreversible.
I want to generate unique random, N-valued key.
This key can contain numbers and latin characters, i.e. A-Za-z0-9.
The only solution I am thinking about is something like this (pseudocode):
key = "";
smb = "ABC…abc…0123456789"; // allowed symbols
for (i = 0; i < N; i++) {
key += smb[rnd(0, smb.length() - 1)]; // select symbol at random position
}
Is there any better solution? What can you suggest?
I would look into GUIDs. From the Wikipedia entry, "the primary purpose of the GUID is to have a totally unique number," which sounds exactly like what you are looking for. There are several implementations out there that generate GUIDs, so it's likely you will not have to reinvent the wheel.
Keeping in mind that the whole field of cryptography relies on, amongst other things, making random numbers. Therefore the NSA, the CIA, and some of the best mathematicians in the world are working on this so I guarantee you that there are better ideas.
Me? I'd just do what fbrereto suggests, and just get a guid. Or look into cryptographic key generators, or y'know, some lava lamps and a camera.
Oh, and as to the code you have; depending on the language, you may need to seed the RNG, or it'll generate the same key every time.
Whatever you do, if you wind up generating a key that uses all numbers and all letters, and if a person is ever going to see that key (which is likely if you are using numbers and letters), omit the characters l, I, 1, O, and 0. People get them confused.
Nothing in your post addresses the question of uniqueness. You're going to have to have some way of not generating the same key twice. Usually, when I need a unique key, I have some unique information to start with. I usually take a one-way hash like MD5, then there are ways to convert that to a key with varying degrees of readability:
Convert to hex
Base64 encode it
Use bits of of the key to index into a list of words.
Example: the unique string computed by hashing the part of this answer above the horizontal line is
abduction's brogue's melted bragger's
You could do a base64 encoding of some random data and remove the +, /, and = characters from the result? I don't know if this would make a predictable distribution. Also, it seems like more work that what you're doing now, which is a fine solution.
Assuming you're using a language/library without an utterly pathetic random number generator, what you've got looks pretty good. N symbols uniformly distributed over a reasonable alphabet works for me, and no amount of applying fancier code is likely to make it more random (just slower).
(For the record, pathetic would include ditching the high-order bits of the underlying random numbers when choosing a value from the given range. While ideally all RNGs would make every bit equally random, in practice that's not so; the higher-order bits tend to be more random. This means that the modulus operator is totally the wrong thing to use when clamping to a restricted range.)
I'd like to know which algorithm is employed. I strongly assume it's something simple and hopefully common. There's no lag in generating the results, for instance.
Input: any string
Output: 5 hex characters (0-F)
I have access to as many keys and results as I wish, but I don't know how exactly I could harness this to attack the function. Is there any method? If I knew any functions that converted to 5-chars to start with then I might be able to brute force for a salt or something.
I know for example that:
a=06a07
b=bfbb5
c=63447
(in case you have something in mind)
In normal use it converts random 32-char strings into 5-char strings.
The only way to derive a hash function from data is through brute force, perhaps combined with some cleverness. There are an infinite number of hash functions, and the good ones perform what is essentially one-way encryption, so it's a question of trial and error.
It's practically irrelevant that your function converts 32-character strings into 5-character hashes; the output is probably truncated. For fun, here are some perfectly legitimate examples, the last 3 of which are cryptographically terrible:
Use the MD5 hashing algorithm, which generates a 16-character hash, and use the 10th through the 14th characters.
Use the SHA-1 algorithm and take the last 5 characters.
If the input string is alphabetic, use the simple substitution A=1, B=2, C=3, ... and take the first 5 digits.
Find each character on your keyboard, measure its distance from the left edge in millimeters, and use every other digit, in reverse order, starting with the last one.
Create a stackoverflow user whose name is the 32-bit string, divide 113 by the corresponding user ID number, and take the first 5 digits after the decimal. (But don't tell 'em I told you to do it!)
Depending on what you need this for, if you have access to as many keys and results as you wish, you might want to try a rainbow table approach. 5 hex chars is only 1mln combinations. You should be able to brute-force generate a map of strings that match all of the resulting hashes in no time. Then you don't need to know the original string, just an equivalent string that generates the same hash, or brute-force entry by iterating over the 1mln input strings.
Following on from a comment I just made to Pontus Gagge, suppose the hash algorithm is as follows:
Append some long, constant string to the input
Compute the SHA-256 hash of the result
Output the last 5 chars of the hash.
Then I'm pretty sure there's no computationally feasible way from your chosen-plaintext attack to figure out what the hashing function is. To even prove that SHA-256 is in use (assuming it's a good hash function, which as far as we currently know it is), I think you'd need to know the long string, which is only stored inside the "black box".
That said, if I knew any published 20-bit hash functions, then I'd be checking those first. But I don't know any: all the usual non-crypto string hashing functions are 32 bit, because that's the expected size of an integer type. You should perhaps compare your results to those of CRC, PJW, and BUZ hash on the same strings, as well as some variants of DJB hash with different primes, and any string hash functions built in to well-known programming languages, like java.lang.String.hashCode. It could be that the 5 output chars are selected from the 8 hex chars generated by one of those.
Beyond that (and any other well-known string hashes you can find), I'm out of ideas. To cryptanalyse a black box hash, you start by looking for correlations between the bits of the input and the bits of the output. This gives you clues what functions might be involved in the hash. But that's a huge subject and not one I'm familiar with.
This sounds mildly illicit.
Not to rain on your parade or anything, but if the implementors have done their work right, you wouldn't notice lags beyond a few tens of milliseconds on modern CPU's even with strong cryptographic hashes, and knowing the algorithm won't help you if they have used salt correctly. If you don't have access to the code or binaries, your only hope is a trivial mistake, whether caused by technical limitations or carelesseness.
There is an uncountable infinity of potential (hash) functions for any given set of inputs and outputs, and if you have no clue better than an upper bound on their computational complexity (from the lag you detect), you have a very long search ahead of you...