On Sat, Jun 07, 2014 at 12:03:29AM +0300, Matti Nykyri wrote: > On Thu, Jun 05, 2014 at 10:58:51PM -0500, Canek Peláez Valdés wrote: > > On Thu, Jun 5, 2014 at 9:56 PM, <meino.cra...@gmx.de> wrote: > > > Hi, > > > > > > I am experimenting with the C code of the ISAAC pseudo random number > > > generator > > > (http://burtleburtle.net/bob/rand/isaacafa.html). > > > > > > Currently the implementation creates (on my embedded linux) 32 bit > > > hexadecimal output. > > > > So it's a 32 bit integer. > > > > > From this I want to create random numbers in the range of [a-Za-z0-9] > > > *without violating randomness* and (if possible) without throwing > > > away bits of the output. > > > > You mean *characters* int the range [A-Za-z0-9]? > > Well this isn't as simple problem as it sounds. A random 32 bit integer > has 32 bits of randomness. If you take a divison reminder of 62 from this > integer you will get only 5,95419631039 bits of randomness > (log(62)/log(2)). So you are wasting 81,4% of your random data. Which is > quite much and usually random data is quite expensive. You can save your > precious random data by taking only 6 bit from your 32 bit integer and > dividing it by 62. Then you will be wasting only 0,8% of random data. > Another problem is alignment, but that is about mathematical correctness. > > > > How can I do this mathemtically (in concern of the quality of output) > > > correct? > > > > The easiest thing to do would be: > > The easiest is not mathematically correct though. Random data will stay > random only if you select and modify it so that randomness is preserved. > If you take devison reminder of 62 from 32 bit integer there are 69 273 > 667 possibilities of the reminder to be 3 or less. For the reminder to 4 > or more the number of possibilities is 69 273 666. In mathematically > ideal case the probability for every index of the list should be same: > 1/62 = 1,61290322581%. But the modulo 62 modifies this probability: for > index 0-3 the probability is 69 273 667/2^32 = 1,61290324759%. And for > indexes 4-61 the probability will be 69 273 666/2^32 = 1,6129032243%. > > If you wish not to waste those random bits the probabilities will get > worse. With 6 bits of random the probability for index 0-1 will be 2/64 > and for 2-63 it will be 1/64. This is a very significant change because > first and second index will appear twice as much as the rest. If you add > 2 characters to your list you will perfect alignment and you can take 6 > bits of data without it modifying probabilities. > > If you are looking a mathematically perfect solution there is a simple > one even if your list is not in the power of 2! Take 6 bits at a time of > the random data. If the result is 62 or 63 you will discard the data and > get the next 6 bits. This selectively modifies the random data but keeps > the probabilities in correct balance. Now the probability for index of > 0-61 is 1/62 because the probability to get 62-63 out of 64 if 0. > > > ------------------------------------------------------------------------------- > > #include <time.h> > > #include <stdio.h> > > #include <stdlib.h> > > > > #define N (26+26+10) > > > > static char S[] = { 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', > > 'K', 'L', 'M', > > 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', > > 'X', 'Y', 'Z', > > 'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j', > > 'k', 'l', 'm', > > 'n', 'o', 'p', 'q', 'r', 's', 't', 'u', 'v', 'w', > > 'x', 'y', 'z', > > '0', '1', '2', '3', '4', '5', '6', '7', '8', '9' }; > > > > int > > next_character() > > { > > // Use the correct call for ISAAC instead of rand() > > unsigned int idx = rand() % N; > > return S[idx]; > > } > > so modify the next_char function: > > char next_character() > { > static unsigned int rand = 0; //(sizeof(int) = 32) > static char bit_avail = 0; > char result = 0; > char move_bits = 0; > char bits_moved = 0; > > do { > if (!bits_avail) { > // Use the correct call for ISAAC instead of rand() > rand = rand(); > > bit_avail = 32; > } > > move_bits = bits_avail >= 6 ? 6 : bits_avail; > result <<= move_bits; > result = (result | rand & (0xFF >> (8 - move_bits))) & 0x3F; > bits_avail -= move_bits; > bits_moved += move_bits; > rand >>= move_bits; > > } while (bits_moved != 6 && result > 61); > > return result; > }
Well actually it looks simpler if you break this like this: unsigned char get_6bits () { static unsigned int rand = 0; //(sizeof(int) = 32) static char bits_avail = 0; unsigned char result = 0; //get 2 bits 3 times: 32 is devidable by 2 for (int i = 0; i < 3; i++) { // --std=c99 //Fill buffer if it is empty! if (!bits_avail || bits_avail < 0 ) { //if bits_avail < 0 it is an error! // Use the correct call for ISAAC instead of rand() rand = rand(); bits_avail = 32; } result <<= 2; //move two bits to left. result = result | (rand & 0x3); //add two least signifigant bits to the result. rand >>= 2; //move two bits to right. bits_avail -= 2; } return result; //result has 6 bits of random data... } char next_character() { unsigned char idx = 0; do { idx = get_6bits(); } while (idx > 61); return S[idx]; } Very simple :) > This function will give perfect distribution of 1/62 probability for > every index. It will waste 6 bits with the probability of 1/32 (2/64). > > > int > > main(int argc, char* argv[]) > > { > > // Use the correct call for initializing the ISAAC seed > > srand((unsigned int)time(NULL)); > > for (int i = 0; i < 20; i++) // --std=c99 > > printf("%c\n", next_character()); > > return 0; > > } > > ------------------------------------------------------------------------------- > > > > If the ISAAC RNG has a good distribution, then the next_character() > > function will give a good distribution among the set [A-Za-z0-9]. > > > > Unless I missunderstood what you meant with "create random numbers in > > the range of [a-Za-z0-9]". -- -Matti