Blowfish Cipher Tutorials - Herong's Tutorial Examples - v2.03, by Herong Yang
piqpr8.c - Generating Hex Digits of PI
A C source code is provided to generate hexadecimal digits of the fractional part of PI (0.1415927...) using the BBP (Bailey–Borwein–Plouffe) formula.
If you are interested in generating hexadecimal digits of the fractional part of PI (0.1415927...) with a C program, here is the source code, piqpr8.c, provided by David H. Bailey using the BBP (Bailey–Borwein–Plouffe) formula:
/* This program implements the BBP algorithm to generate a few hexadecimal digits beginning immediately after a given position id, or in other words beginning at position id + 1. On most systems using IEEE 64-bit floating-point arithmetic, this code works correctly so long as d is less than approximately 1.18 x 10^7. If 80-bit arithmetic can be employed, this limit is significantly higher. Whatever arithmetic is used, results for a given position id can be checked by repeating with id-1 or id+1, and verifying that the hex digits perfectly overlap with an offset of one, except possibly for a few trailing digits. The resulting fractions are typically accurate to at least 11 decimal digits, and to at least 9 hex digits. */ /* David H. Bailey 2006-09-08 */ #include <stdio.h> #include <math.h> main() { double pid, s1, s2, s3, s4; double series (int m, int n); void ihex (double x, int m, char c[]); int id = 1000000; #define NHX 16 char chx[NHX]; /* id is the digit position. Digits generated follow immediately after id. */ s1 = series (1, id); s2 = series (4, id); s3 = series (5, id); s4 = series (6, id); pid = 4. * s1 - 2. * s2 - s3 - s4; pid = pid - (int) pid + 1.; ihex (pid, NHX, chx); printf(" position = %i\n fraction = %.15f \n hex digits = %10.10s\n", id, pid, chx); } void ihex (double x, int nhx, char chx[]) /* This returns, in chx, the first nhx hex digits of the fraction of x. */ { int i; double y; char hx[] = "0123456789ABCDEF"; y = fabs (x); for (i = 0; i < nhx; i++){ y = 16. * (y - floor (y)); chx[i] = hx[(int) y]; } } double series (int m, int id) /* This routine evaluates the series sum_k 16^(id-k)/(8*k+m) using the modular exponentiation technique. */ { int k; double ak, eps, p, s, t; double expm (double x, double y); #define eps 1e-17 s = 0.; /* Sum the series up to id. */ for (k = 0; k < id; k++){ ak = 8 * k + m; p = id - k; t = expm (p, ak); s = s + t / ak; s = s - (int) s; } /* Compute a few terms where k >= id. */ for (k = id; k <= id + 100; k++){ ak = 8 * k + m; t = pow (16., (double) (id - k)) / ak; if (t < eps) break; s = s + t; s = s - (int) s; } return s; } double expm (double p, double ak) /* expm = 16^p mod ak. This routine uses the left-to-right binary exponentiation scheme. */ { int i, j; double p1, pt, r; #define ntp 25 static double tp[ntp]; static int tp1 = 0; /* If this is the first call to expm, fill the power of two table tp. */ if (tp1 == 0) { tp1 = 1; tp[0] = 1.; for (i = 1; i < ntp; i++) tp[i] = 2. * tp[i-1]; } if (ak == 1.) return 0.; /* Find the greatest power of two less than or equal to p. */ for (i = 0; i < ntp; i++) if (tp[i] > p) break; pt = tp[i-1]; p1 = p; r = 1.; /* Perform binary exponentiation algorithm modulo ak. */ for (j = 1; j <= i; j++){ if (p1 >= pt){ r = 16. * r; r = r - (int) (r / ak) * ak; p1 = p1 - pt; } pt = 0.5 * pt; if (pt >= 1.){ r = r * r; r = r - (int) (r / ak) * ak; } } return r; }
For more details on piqpr8.c, please see "BBP Code Directory" at http://www.experimentalmath.info/bbp-codes/.
Of course, you can also read the Wikipedia article "Bailey–Borwein–Plouffe formula" at https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula.
Table of Contents
Blowfish Cipher Encryption Algorithm
Key Schedule (Sub-Keys and S-Boxes) Algorithm
Efficient Form of the Blowfish Algorithm
Blowfish Cipher Decryption Algorithm
Proof of Blowfish Cipher Algorithm
First 8336 Fractional Hex Digits of PI
►piqpr8.c - Generating Hex Digits of PI
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