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Author: scha00

Miller-Rabin Primality Test/Images/img_pseudo.png

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+# Miller-Rabin Primality Test
+
+In 1976, Gray Miller introduced an algorithm, through his ph.d thesis[1], which determines a primality of the given number. The original algorithm was deterministic under the Extended Reimann Hypothesis, which is yet to be proven. After four years, Michael O. Rabin improved the algorithm[2] by using probabilistic approach and it no longer assumes the unproven hypothesis.
+
+## Probabilistic
+
+The result of the test is simply a boolean. However, `true` does not implicate *the number is prime*. In fact, it means *the number is **probably** prime*. But `false` does mean *the number is composite*.
+
+In order to increase the accuracy of the test, it needs to be iterated few times. If it returns `true` in every single iteration, than we can say with confidence that *the number is pro......bably prime*.
+
+## Algorithm
+
+Let `n` be the given number, and write `n-1` as `2^s·d`, where d is odd. And choose a random number `a` within the range from `2` to `n - 1`.
+
+Now make a sequence, in modulo `n`, as following:
+
+> a^d, a^(2·d), a^(4·d), ... , a^((2^(s-1))·d), a^((2^s)·d) = a^(n-1)
+
+And we say the number `n` passes the test, *probably prime*, if 1) `a^d` is congruence to `1` in modulo `n`, or 2) `a^((2^k)·d)` is congruence to `-1` for some `k = 1, 2, ..., s-1`.
+
+### Pseudo Code
+
+The following pseudo code is excerpted from Wikipedia[3]:
+
+![Image of Pseudocode](./Images/img_pseudo.png)
+
+## Usage
+
+```swift
+mrPrimalityTest(7)                      // test if 7 is prime. (default iteration = 1)
+mrPrimalityTest(7, iteration: 10)       // test if 7 is prime & iterate 10 times.
+```
+
+## Reference
+1. G. L. Miller, "Riemann's Hypothesis and Tests for Primality". *J. Comput. System Sci.* 13 (1976), 300-317.
+2. M. O. Rabin, "Probabilistic algorithm for testing primality". *Journal of Number Theory.* 12 (1980), 128-138.
+3. Miller–Rabin primality test - Wikipedia, the free encyclopedia
+
+*Written for Swift Algorithm Club by **Sahn Cha**, @scha00*
+
+[1]: https://cs.uwaterloo.ca/research/tr/1975/CS-75-27.pdf
+[2]: http://www.sciencedirect.com/science/article/pii/0022314X80900840
+[3]: https://en.wikipedia.org/wiki/Miller–Rabin_primality_test