Add description of exponentiation by factoring.

hselasky · adamant-pwn · commit 187a4261dea5 · 2022-05-12T01:01:13.000+02:00
diff --git a/src/algebra/factoring-exp.md b/src/algebra/factoring-exp.md
@@ -0,0 +1,113 @@
+# Binary Exponentiation by Factoring
+
+Binary exponentiation by factoring is a binary trick (*) which allows to
+calculate $a^n$ using only $O(1)$ multiplications, instead of $O(log n)$
+multiplications required by the classical binary exponentiation approach.
+
+## Algorithm
+
+Raising $a$ to the power of $n$ is expressed naively as the exponent
+of the logarithm of $a$ multiplied by $n$ being the exponent.  In the
+following examples all base numbers, $a$, are assumed to be odd.
+
+## Implementation for 32-bit integers
+
+### The logarithm table needed
+
+```cpp
+static const uint32_t mbin_log_32_table[32] = {
+        0x00000000,
+        0x00000000,
+        0xd3cfd984,
+        0x9ee62e18,
+        0xe83d9070,
+        0xb59e81e0,
+        0xa17407c0,
+        0xce601f80,
+        0xf4807f00,
+        0xe701fe00,
+        0xbe07fc00,
+        0xfc1ff800,
+        0xf87ff000,
+        0xf1ffe000,
+        0xe7ffc000,
+        0xdfff8000,
+        0xffff0000,
+        0xfffe0000,
+        0xfffc0000,
+        0xfff80000,
+        0xfff00000,
+        0xffe00000,
+        0xffc00000,
+        0xff800000,
+        0xff000000,
+        0xfe000000,
+        0xfc000000,
+        0xf8000000,
+        0xf0000000,
+        0xe0000000,
+        0xc0000000,
+        0x80000000,
+};
+```
+
+### The logarithmic function
+
+```cpp
+static uint32_t
+mbin_log_32(uint32_t r, uint32_t x)
+{
+        uint8_t n;
+
+        for (n = 2; n != 16; n++) {
+                if (x & (1 << n)) {
+                        x = x + (x << n);
+                        r -= mbin_log_32_table[n];
+                }
+        }
+
+        r -= (x & 0xFFFF0000);
+
+        return (r);
+}
+```
+
+### The exponent function
+
+```cpp
+static uint32_t
+mbin_exp_32(uint32_t r, uint32_t x)
+{
+        uint8_t n;
+
+        for (n = 2; n != 16; n++) {
+                if (x & (1 << n)) {
+                        r = r + (r << n);
+                        x -= mbin_log_32_table[n];
+                }
+        }
+
+        r *= 1 - (x & 0xFFFF0000);
+
+        return (r);
+}
+```
+
+### The exponentiation function
+
+```cpp
+static uint32_t
+mbin_power_odd_32(uint32_t rem, uint32_t base, uint32_t exp)
+{
+        if (base & 2) {
+                /* divider is considered negative */
+                base = -base;
+                /* check if result should be negative */
+                if (exp & 1)
+                        rem = -rem;
+        }
+        return (mbin_exp_32(rem, mbin_log_32(0, base) * exp));
+}
+```
+
+* [M30, Hans Petter Selasky, 2009](https://books.google.no/books?id=IkuEBAAAQBAJ&pg=PT310&lpg=PT310&dq=M30+selasky&source=bl&ots=W86b0P-yfw&sig=ACfU3U0Z7orvRJyZSGuUbOQ8H85sSpZHpQ&hl=no&sa=X&ved=2ahUKEwiC87OO1Zn3AhXrtYsKHTcIAgUQ6AF6BAgCEAM#v=onepage&q=M30%20selasky&f=false)
diff --git a/src/index.md b/src/index.md
@@ -33,6 +33,7 @@ Full list of updates: [Commit History](https://github.com/e-maxx-eng/e-maxx-eng/
 
 - **Fundamentals**
     - [Binary Exponentiation](algebra/binary-exp.md)
+    - [Factoring Exponentiation](algebra/factoring-exp.md)
     - [Euclidean algorithm for computing the greatest common divisor](algebra/euclid-algorithm.md)
     - [Extended Euclidean Algorithm](algebra/extended-euclid-algorithm.md)
     - [Linear Diophantine Equations](algebra/linear-diophantine-equation.md)
