clarify comment in PSOV2Encryption::single

src/PSOEncryption.cc

@@ -111,7 +111,9 @@ uint32_t PSOV2Encryption::single(uint32_t seed) {
   // If fib(n) is the n'th Fibonacci number (starting with 1, 1, 2, 3, 5, etc.), then a closed form for the integer
   // sequence generated by the first loop in PSOV2Encryption::PSOV2Encryption is:
   //   a(n) = (-1)^n * (fib(n) - fib(n-1) * seed)
-  // The recurrence used in that loop is a(n) = a(n-2) - a(n-1), which we can use to prove the closed form correct:
+  // The sequence begins with a(-1) = seed (which is not generated by the loop but is used as an initial input, hence
+  // the negative index) and a(0) = 1, and the recurrence used in that loop is a(n) = a(n-2) - a(n-1). Assuming that
+  // a(n-2) and a(n-1) are described by this closed form, we can show that a(n) is as well:
   //   a(n) = a(n-2) - a(n-1)
   //   a(n) = (-1)^(n-2) * (fib(n-2) - fib(n-3) * seed) - ((-1)^(n-1) * (fib(n-1) - fib(n-2) * seed))
   //   a(n) = (-1)^(n-2) * (fib(n-2) - fib(n-3) * seed) + ((-1)^(n-2) * (fib(n-1) - fib(n-2) * seed))
@@ -119,13 +121,13 @@ uint32_t PSOV2Encryption::single(uint32_t seed) {
   //   a(n) = (-1)^(n-2) * (fib(n-2) + fib(n-1) - (fib(n-3) + fib(n-2)) * seed)
   //   a(n) = (-1)^(n-2) * (fib(n) - fib(n-1) * seed)
   //   a(n) = (-1)^(n) * (fib(n) - fib(n-1) * seed)
-  // The sequence begins with a(-1) = seed (which is not generated by the loop but is used as an initial input, hence
+  // This shows inductively that this closed form holds for all n >= 1 (since the sequence begins with a(-1)). Using
-  // the negative index) and a(0) = 1. Using the closed form and the values of a(-1) and a(0), we can eliminate all
+  // the closed form and the values of a(-1) and a(0), we can eliminate all arithmetic done in the normal constructor
-  // arithmetic done in the normal constructor that isn't necessary to produce the first result value. To do so, we
+  // that isn't necessary to produce the first result value. To do so, we trace backward from the result value, through
-  // trace backward from the result value, through the 5 update_stream calls and the initialization loop, to see which
+  // the 5 update_stream calls and the initialization loop, to see which indexes within the stream are actually needed,
-  // indexes within the stream are actually needed, and the expression to generate each one. We can then simplify the
+  // and the expression to generate each one. We can then simplify the overall expression and truncate constants to 32
-  // overall expression and truncate constants to 32 bits (since it's a linear equation, overflow bits cannot affect
+  // bits (since it's a linear equation, overflow bits cannot affect the final 32-bit result). The full expression
-  // the final 32-bit result). The full expression simplifies to:
+  // simplifies to:
   return 0xC6DCAB76 * seed - 0x9E1977BA;
 }