I want to generate using the middle square method 10,000 (ten thousand) numbers with 6 decimals for both higher than 1 (for example 785633)and lower than 1(for example 0.434367) starting numbers. Is there any starting number for the two situations that can generate 10,000 distinct numbers?
You generally want a pretty big number for middle-square, say fifty digits or so. When you pick six digits (they can be any portion of the middle part of the number), you can use them as a six-digit number or divide by a million and use them as a decimal number.
You should be aware that middle-square is no longer considered a good method for generating random numbers. A simple linear congruential generator is faster and better, and there are many other types of random number generators also.
Related
I want a simple (non-cryptographic) random number generation algorithm where I can freely choose the period.
One candidate would be a special instance of LCG:
X(n+1) = (aX(n)+c) mod m (m,c relatively prime; (a-1) divisible by all prime factors of m and also divisible by 4 if m is).
This has period m and does not restrict possible values of m.
I intend to use this RNG to create a permutation of an array by generating indices into it. I tried the LCG and it might be OK. However, it may not be "random enough" in that distances between adjacent outputs have very few possible values (i.e, plotting x(n) vs n gives a wrapped line). The arrays I want to index into have some structure that has to do with this distance and I want to avoid potential issues with this.
Of course, I could use any good PRNG to shuffle (using e.g. Fisher–Yates) an array [1,..., m]. But I don't want to have to store this array of indices. Is there some way to capture the permuted indices directly in an algorithm?
I don't really mind the method ending up biased w.r.t choice of RNG seed. Only the period matters and the permuted sequence (for a given seed) being reasonably random.
Encryption is a one-to-one operation. If you encrypt a range of numbers, you will get the same count of apparently random numbers back. In this case the period will be the size of the chosen range. So for a period of 20, encrypt the numbers 0..19.
If you want the output numbers to be in a specific range, then pick a block cipher with an appropriately sized block and use Format Preserving Encryption if needed, as #David Eisenstat suggests.
It is not difficult to set up a cipher with almost any reasonable block size, so long as it is an even number of bits, using the Feistel structure. If you don't require cryptographic security then four or six Feistel rounds should give you enough randomness.
Changing the encryption key will give you a different ordering of the numbers.
I am looking for a shuffle algorithm to shuffle a set of sequential numbers without buffering. Another way to state this is that I’m looking for a random sequence of unique numbers that have a given period.
Your typical Fisher–Yates shuffle needs to have each element all of the elements it is going to shuffle, so that isn’t going to work.
A Linear-Feedback Shift Register (LFSR) does what I want, but only works for periods that are powers-of-two less two. Here is an example of using a 4-bit LFSR to shuffle the numbers 1-14:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
8
12
14
7
4
10
5
11
6
3
2
1
9
13
The first two is the input, and the second row the output. What’s nice is that the state is very small—just the current index. You can start of any index and get a difference set of numbers (starting at 1 yields: 8, 12, 14; starting at 9: 6, 3, 2), although the sequence is always the same (5 is always followed by 11). If I want a different sequence, I can pick a different generator polynomial.
The limitations to the LFSR are that the periods are always power-of-two less two (the min and max are always the same, thus unshuffled) and there not enough enough generator polynomials to allow every possible random sequence.
A block cipher algorithm would work. Every key produces a uniquely shuffled set of numbers. However all block ciphers (that I know about) have power-of-two block sizes, and usually a fixed or limited number of block sizes. A block cipher with a arbitrary non-binary block size would be perfect if such a thing exists.
There are a couple of projects I have that could benefit from such an algorithm. One is for small embedded micros that need to produce a shuffled sequence of numbers with a period larger than the memory they have available (think Arduino Uno needing to shuffle 1 to 100,000).
Does such an algorithm exist? If not, what things might I search for to help me develop such an algorithm? Or is this simply not possible?
Edit 2022-01-30
I have received a lot of good feedback and I need to better explain what I am searching for.
In addition to the Arduino example, where memory is an issue, there is also the shuffle of a large number of records (billions to trillions). The desire is to have a shuffle applied to these records without needing a buffer to hold the shuffle order array, or the time needed to build that array.
I do not need an algorithm that could produce every possible permutation, but a large number of permutations. Something like a typical block cipher in counter mode where each key produces a unique sequence of values.
A Linear Congruential Generator using coefficients to produce the desired sequence period will only produce a single sequence. This is the same problem for a Linear Feedback Shift Register.
Format-Preserving Encryption (FPE), such as AES FFX, shows promise and is where I am currently focusing my attention. Additional feedback welcome.
It is certainly not possible to produce an algorithm which could potentially generate every possible sequence of length N with less than N (log2N - 1.45) bits of state, because there are N! possible sequence and each state can generate exactly one sequence. If your hypothetical Arduino application could produce every possible sequence of 100,000 numbers, it would require at least 1,516,705 bits of state, a bit more than 185Kib, which is probably more memory than you want to devote to the problem [Note 1].
That's also a lot more memory than you would need for the shuffle buffer; that's because the PRNG driving the shuffle algorithm also doesn't have enough state to come close to being able to generate every possible sequence. It can't generate more different sequences than the number of different possible states that it has.
So you have to make some compromise :-)
One simple algorithm is to start with some parametrisable generator which can produce non-repeating sequences for a large variety of block sizes. Then you just choose a block size which is as least as large as your target range but not "too much larger"; say, less than twice as large. Then you just select a subrange of the block size and start generating numbers. If the generated number is inside the subrange, you return its offset; if not, you throw it away and generate another number. If the generator's range is less than twice the desired range, then you will throw away less than half of the generated values and producing the next element in the sequence will be amortised O(1). In theory, it might take a long time to generate an individual value, but that's not very likely, and if you use a not-very-good PRNG like a linear congruential generator, you can make it very unlikely indeed by restricting the possible generator parameters.
For LCGs you have a couple of possibilities. You could use a power-of-two modulus, with an odd offset and a multiplier which is 5 mod 8 (and not too far from the square root of the block size), or you could use a prime modulus with almost arbitrary offset and multiplier. Using a prime modulus is computationally more expensive but the deficiencies of LCG are less apparent. Since you don't need to handle arbitrary primes, you can preselect a geometrically-spaced sample and compute the efficient division-by-multiplication algorithm for each one.
Since you're free to use any subrange of the generator's range, you have an additional potential parameter: the offset of the start of the subrange. (Or even offsets, since the subrange doesn't need to be contiguous.) You can also increase the apparent randomness by doing any bijective transformation (XOR/rotates are good, if you're using a power-of-two block size.)
Depending on your application, there are known algorithms to produce block ciphers for subword bit lengths [Note 2], which gives you another possible way to increase randomness and/or add some more bits to the generator state.
Notes
The approximation for the minimum number of states comes directly from Stirling's approximation for N!, but I computed the number of bits by using the commonly available lgamma function.
With about 30 seconds of googling, I found this paper on researchgate.net; I'm far from knowledgable enough in crypto to offer an opinion, but it looks credible; also, there are references to other algorithms in its footnotes.
I'm having difficulty storing and displaying numbers greater than 32767 in LC-3 since a register can only hold values from -32768 to 32767. My apology for not being able to come up with any idea for the algorithm. Please give me some suggestion. Thanks!
You'll need a representation to store the larger number in a pair or more of words.
There are several approaches to how big integers are stored: in a fixed number of words, and in a variable number of words or bytes. The critical part is being able to detect the presence and amount of overflow/carry on mathematical operations like *10.
For that reason, one simple approach is to use a variable number of words/bytes (for a single number), and store only one decimal digit in each of the words/bytes. That way multiplication by 10, means simply adding a digit on the end (which has the effect of moving each existing digit to the next higher power of ten position). Adding numbers of this form numbers is fairly easy as well, we need to line up the digits and then, we add them up and detect when the sum is >= 10, then there is a carry (of 1) to be added to the next higher order digit of the sum. (If adding two such (variable length) numbers is desired, I would store the decimal digits in reverse order, because then the low order numbers are already lined up for addition.) See also https://en.wikipedia.org/wiki/Binary-coded_decimal . (In some sense, this is like storing numbers in a form like string, but using binary values instead of ascii characters.)
To simplify this approach for your needs, you can fix the number of words to use, e.g. at 7, for 7 digits.
A variation on (unpacked) Binary-coded Decimal to pack them two decimal digits per byte. Its a bit more complicated but saves some storage.
Another approach is to store as many decimal digits as will fit full in a word, minus 1. Which is to say if we can store 65536 in 16-bits that's only 4 full decimal digits, which means putting 3 digits at a time into a word. You'd need 3 words for 9 digits. Multiplication by 10 means multiplying each word by 10 numerically, and then checking for larger than 999, and if larger, then carry the 1 to the next higher order word while also subtracting 10,000 from the overflowing word.
This approach will require actual multiplication and division by 10 on each of the individual words.
There are other approaches, such as using all 16-bits in a word as magnitude, but the difficulty there is determining the amount of overflow/carry on *10 operations. It is not a monumental task but will require work. See https://stackoverflow.com/a/1815371/471129, for example.
(If you also want to store negative numbers, that is also an issue for representation. We can either store the sign as separately known as sign-magnitude form (as in stored its own word/byte or packed into the highest byte) or store the number in a compliment form. The former is better for variable length implementations and the latter can be made to work for fixed length implementations.)
I'm trying to make this algorithm which inputs a lower and upper limit for two numbers (the two numbers may have different lower and upper limits) and outputs two random numbers within that range
The catch is however that when the two numbers are added, no "carry" should be there. This means the sum of the digits in each place should be no more than 9.
How can I make sure that the numbers are truly random and that no carrying occurs when adding the two numbers
Thanks a lot!
Edit: The ranges can vary, the widest range can be 0 to 999. Also, I'm using VBA (Excel)
An easy and distributionally correct way of doing this is to use Rejection Sampling, a.k.a. "Acceptance/Rejection". Generate the values independently, and if the carry constraint is violated repeat. In pseudocode
do {
generate x, y
} while (x + y > threshold)
The number of times the loop will iterate has a geometric distribution with an expected value of (proportion of sums below the threshold)-1. For example, if you're below the threshold 90% of the time then the long term number of iterations will average out to 10/9, 1.11... iterations per pair generated. For lower likelihoods of acceptance, it will take more attempts on average.
I have a set of 64-bit unsigned integers with length >= 2. I pick 2 random integers, a, b from that set. I apply a deterministic operation to combine a and b into different 64-bit unsigned integers, c_1, c_2, c_3, etc. I add those c_ns to the set. I repeat that process.
What procedure can I use to guarantee that c will practically never collide with an existing bitstring on the set, even after millions of steps?
Since you're generating multiple 64-bit values from a pair of 64-bit numbers, I would suggest that you select two numbers at random, and use them to initialize a 64 bit xorshift random number generator with 128 bits of state. See https://en.wikipedia.org/wiki/Xorshift#xorshift.2B for an example.
However, it's rather difficult to predict the collision probability when you're using multiple random number generators. With a single PRNG, the rule of thumb is that you'll have a 50% chance of a collision after generating the square root of the range. For example, if you were generating 32-bit random numbers, your collision probability reaches 50% after about 70,000 numbers generated. Square root of 2^32 is 65,536.
With a single 64-bit PRNG, you could generate more than a billion random numbers without too much worry about collisions. In your case, you're picking two numbers from a potentially small pool, then initializing a PRNG and generating a relatively small number of values that you add back to the pool. I don't know how to calculate the collision probability in that case.
Note, however, that whatever the probability of collision, the possibility of collision always exists. That "one in a billion" chance does in fact occur: on average once every billion times you run the program. You're much better off saving your output numbers in a hash set or other data structure that won't allow you to store duplicates.
I think the best you can do without any other given constraints is to use a pseudo-random function that maps two 64-bit integers to a 64-bit integer. Depending on whether the order of a and b matter for your problem or not (i.e. (3, 5) should map to something else than (5, 3)) you shouldn't or should sort them before.
The natural choice for a pseudo-random function that maps a larger input to a smaller input is a hash function. You can select any hash function that produces an output of at least 64-bit and truncate it. (My favorite in this case would be SipHash with an arbitrary fixed key, it is fast and has public domain implementations in many languages, but you might just use whatever is available.)
The expected amount of numbers you can generate before you get a collision is determined by the birthday bound, as you are essentially selecting values at random. The linked article contains a table for the probabilities for 64-bit values. As an example, if you generate about 6 million entries, you have a collision probability of one in a million.
I don't think it is possible to beat this approach in the general case, as you could encode an arbitrary amount of information in the sequence of elements you combine while the amount of information in the output value is fixed to 64-bit. Thus you have to consider collisions, and a random function spreads out the probability evenly among all possible sequences.