rand uniform: add comments outlining the algorithm

Reviewed-by: Tom Cosgrove <[email protected]>
Reviewed-by: Matthias St. Pierre <[email protected]>
Reviewed-by: Tomas Mraz <[email protected]>
(Merged from openssl/openssl#22499)

Signed-off-by: fly2x <[email protected]>

Author: paulidale

crypto/rand/rand_uniform.c

@@ -12,7 +12,14 @@
 
 /*
  * Implementation an optimal random integer in a range function.
- * Refer: https://github.com/apple/swift/pull/39143 for a description
+ *
+ * Essentially it boils down to incrementally generating a fixed point
+ * number on the interval [0, 1) and multiplying this number by the upper
+ * range limit.  Once it is certain what the fractional part contributes to
+ * the integral part of the product, the algorithm has produced a definitive
+ * result.
+ *
+ * Refer: https://github.com/apple/swift/pull/39143 for a fuller description
  * of the algorithm.
  */
 uint32_t ossl_rand_uniform_uint32(OSSL_LIB_CTX *ctx, uint32_t upper, int *err)
@@ -29,17 +36,43 @@ uint32_t ossl_rand_uniform_uint32(OSSL_LIB_CTX *ctx, uint32_t upper, int *err)
     }
     if (unlikely(upper == 1))
         return 0;
+
     /* Get 32 bits of entropy */
     if (RAND_bytes_ex(ctx, (unsigned char *)&rand, sizeof(rand), 0) <= 0) {
         *err = 1;
         return 0;
     }
+
+    /*
+     * We are generating a fixed point number on the interval [0, 1).
+     * Multiplying this by the range gives us a number on [0, upper).
+     * The high word of the multiplication result represents the integral
+     * part we want.  The lower word is the fractional part.  We can early exit if
+     * if the fractional part is small enough that no carry from the next lower
+     * word can cause an overflow and carry into the integer part.  This
+     * happens when the fractional part is bounded by 2^32 - upper which
+     * can be simplified to just -upper (as an unsigned integer).
+     */
     prod = (uint64_t)upper * rand;
     i = prod >> 32;
     f = prod & 0xffffffff;
     if (likely(f <= 1 + ~upper))    /* 1+~upper == -upper but compilers whine */
         return i;
 
+    /*
+     * We're in the position where the carry from the next word *might* cause
+     * a carry to the integral part.  The process here is to generate the next
+     * word, multiply it by the range and add that to the current word.  If
+     * it overflows, the carry propagates to the integer part (return i+1).
+     * If it can no longer overflow regardless of further lower order bits,
+     * we are done (return i).  If there is still a chance of overflow, we
+     * repeat the process with the next lower word.
+     *
+     * Each *bit* of randomness has a probability of one half of terminating
+     * this process, so each each word beyond the first has a probability
+     * of 2^-32 of not terminating the process.  That is, we're extremely
+     * likely to stop very rapidly.
+     */
     for (j = 0; j < max_followup_iterations; j++) {
         if (RAND_bytes_ex(ctx, (unsigned char *)&rand, sizeof(rand), 0) <= 0) {
             *err = 1;
@@ -59,8 +92,8 @@ uint32_t ossl_rand_uniform_uint32(OSSL_LIB_CTX *ctx, uint32_t upper, int *err)
     }
     /*
      * If we get here, we've consumed 32 * max_followup_iterations + 32 bits
-     * with no firm decision, this gives a bias with probability < 2^(32*n),
-     * likely acceptable.
+     * with no firm decision, this gives a bias with probability < 2^-(32*n),
+     * which is likely acceptable.
      */
     return i;
 }