Miller Rabin Primality Test Calculator
Test if large numbers are prime using the Miller-Rabin probabilistic primality test. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Miller Rabin Primality Test Calculator
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Formula: Test: a^d mod n and a^(2^i * d) mod n for i = 0 to r-1
Worked example — Probably Prime | Confidence: 99.9999% | 10 iterations
Formula
Test: a^d mod n and a^(2^i * d) mod n for i = 0 to r-1
The Miller-Rabin test decomposes n-1 as 2^r * d (d odd), then for each test base a, checks if a^d mod n = 1 or if any a^(2^i * d) mod n = n-1. If neither holds, n is composite. Each test reduces the error probability by a factor of 4.
Worked Examples
Example 1: Testing 104729 for Primality
Problem:Use the Miller-Rabin test with 10 iterations to determine if 104729 is prime.
Solution:n = 104729, n-1 = 104728 Decompose: 104728 = 2^3 x 13091, so r = 3, d = 13091 Test base 2: 2^13091 mod 104729 = check sequence Test base 3: 3^13091 mod 104729 = check sequence ... (continue for remaining bases) All bases pass the primality conditions. Error bound: (1/4)^10 = 9.5367e-7 Confidence: 99.9999%
Result:Probably Prime | Confidence: 99.9999% | 10 iterations
Example 2: Testing 561 (Carmichael Number)
Problem:Test 561, the smallest Carmichael number, using Miller-Rabin.
Solution:n = 561, n-1 = 560 Decompose: 560 = 2^4 x 35, so r = 4, d = 35 Trial division: 561 / 3 = 187 (factor found!) 561 = 3 x 11 x 17 Note: 561 passes the Fermat test for all coprime bases but Miller-Rabin detects it as composite. Fermat test alone would incorrectly identify it as prime.
Result:Composite | Factor found: 3 | 561 = 3 x 187
Frequently Asked Questions
What is the Miller-Rabin primality test and how does it work?
The Miller-Rabin primality test is a probabilistic algorithm that determines whether a given number is likely prime or definitely composite. It is based on properties of modular arithmetic related to Fermat's Little Theorem. The algorithm works by decomposing n-1 into the form 2^r * d where d is odd, then testing random bases a to see if a^d mod n equals 1 or if any a^(2^i * d) mod n equals n-1 for some i between 0 and r-1. If neither condition holds for a particular base, the number is definitely composite and that base is called a witness to compositeness. If both conditions hold for all tested bases, the number is probably prime. Each iteration reduces the probability of a false positive by a factor of 4.
How reliable is the Miller-Rabin test and what is the probability of error?
The Miller-Rabin test has a worst-case error probability of at most 1/4 per iteration, meaning each additional random base tested reduces the probability of incorrectly declaring a composite number as prime by a factor of 4. With k iterations, the error probability is at most (1/4)^k. For 10 iterations, the error probability is less than 1 in a million. For 20 iterations, it drops below 1 in a trillion. In practice, the actual error rate is much lower than this theoretical bound because most composite numbers have many witnesses. The test never produces false negatives: if it says a number is composite, that determination is always correct. Only the probably prime result carries uncertainty.
What is the difference between deterministic and probabilistic primality testing?
Deterministic primality tests, such as the AKS algorithm discovered in 2002, produce a guaranteed correct answer for any input with zero probability of error. However, deterministic tests are generally slower for very large numbers. Probabilistic tests like Miller-Rabin trade absolute certainty for dramatically faster execution. The Miller-Rabin test runs in O(k * log^2(n)) time, making it practical for numbers with hundreds of digits. For numbers below certain thresholds, specific sets of bases make Miller-Rabin deterministic: testing bases 2, 3, 5, and 7 gives correct results for all numbers below 3,215,031,751. The practical reality is that probabilistic tests with sufficient iterations are reliable enough for cryptographic applications where they are used to generate RSA key pairs.
Why is primality testing important for cryptography and computer security?
Modern public-key cryptography, particularly the RSA algorithm, relies fundamentally on the difficulty of factoring large numbers that are the product of two large primes. Generating RSA keys requires finding two large prime numbers, typically 512 to 2048 bits each. The Miller-Rabin test is the industry-standard method for this because it can quickly verify primality of numbers with hundreds of digits. If primality testing were unreliable, cryptographic keys could be generated from non-prime numbers, making them trivially breakable. The security of internet banking, encrypted communications, and digital signatures all depend on fast, reliable primality testing. Most cryptographic libraries use 40 or more iterations of Miller-Rabin, making the error probability astronomically small.
What are witnesses and liars in the context of Miller-Rabin testing?
In Miller-Rabin terminology, a witness is a base value a that proves a number is composite. If testing base a against number n shows that n fails the primality conditions, then a is called a witness to the compositeness of n. A liar is a base that makes a composite number appear to pass the primality test. For any composite odd number n, at most 1/4 of all possible bases between 2 and n-2 are liars, which is why the error probability per iteration is at most 25 percent. Strong pseudoprimes are composite numbers that have many liars, making them harder to detect as composite. The smallest strong pseudoprime to base 2 is 2047, and it requires testing with base 3 as well to correctly identify it as composite.
How does the decomposition of n-1 into 2^r * d work in the Miller-Rabin test?
The decomposition of n-1 into the form 2^r * d where d is odd is a crucial first step of the Miller-Rabin algorithm. Since n is odd and greater than 2, n-1 is even and can be repeatedly divided by 2. The exponent r counts how many times n-1 is divisible by 2, and d is the remaining odd factor. For example, if n = 221, then n-1 = 220 = 4 * 55 = 2^2 * 55, giving r = 2 and d = 55. This decomposition allows the algorithm to check a sequence of modular exponentiations: a^d, a^(2d), a^(4d), up to a^(2^(r-1) * d). These values form a chain where each is the square of the previous one modulo n, and the pattern of these values reveals whether n is prime or composite.
What is the relationship between Fermat's Little Theorem and the Miller-Rabin test?
The Miller-Rabin test is a strengthened version of the Fermat primality test, which is based on Fermat's Little Theorem stating that if p is prime and a is not divisible by p, then a^(p-1) mod p = 1. The Fermat test simply checks whether a^(n-1) mod n equals 1, but it fails for Carmichael numbers, which are composite numbers that pass the Fermat test for all bases coprime to them. The smallest Carmichael number is 561. Miller-Rabin strengthens this by examining the intermediate values during the computation of a^(n-1) mod n, specifically looking at the square root chain derived from the 2^r * d decomposition. This additional analysis catches Carmichael numbers and dramatically reduces false positives compared to the basic Fermat test.
How do you perform modular exponentiation efficiently for large numbers?
Modular exponentiation, computing a^b mod m efficiently, uses the method of repeated squaring, also called binary exponentiation or exponentiation by squaring. Instead of computing a^b directly (which would produce astronomically large intermediate values), the algorithm processes the binary representation of the exponent b from right to left. At each step, if the current bit of b is 1, multiply the running result by the current power of a (all modulo m). Then square the current power of a for the next bit. This reduces the number of multiplications from b to approximately 2 * log2(b), making it feasible to compute modular powers of numbers with hundreds of digits. For example, computing 3^100 mod 7 requires only about 14 multiplications instead of 100.
Can the Miller-Rabin test find the actual factors of a composite number?
The Miller-Rabin test is designed only to determine whether a number is prime or composite; it does not find factors. When it identifies a composite number, it provides a witness base but not the actual divisors. However, in some cases, the intermediate computations during the test can incidentally reveal a factor. Specifically, if during the squaring sequence a value x is found where x^2 mod n = 1 but x is neither 1 nor n-1, then gcd(x-1, n) or gcd(x+1, n) will yield a non-trivial factor of n. This is the basis of some factoring algorithms. For dedicated factoring, algorithms like Pollard rho, quadratic sieve, or the general number field sieve are used, which are specifically designed to find the prime factors of composite numbers.
What numbers are commonly used as test bases in practical Miller-Rabin implementations?
In practical implementations, specific deterministic base sets are used for numbers within certain ranges to guarantee correct results without any probability of error. For numbers below 2,047, testing with base 2 alone is sufficient. For numbers below 1,373,653, bases 2 and 3 suffice. For numbers below 3,215,031,751, bases 2, 3, 5, and 7 give deterministic results. For numbers below 3,317,044,064,679,887,385,961,981, the first 12 prime bases (2 through 37) are sufficient. For arbitrary large numbers used in cryptography, random bases are chosen and 40 to 64 iterations are typical. The OpenSSL library uses a variable number of iterations based on key size, typically 56 iterations for 512-bit numbers and 6 iterations for 2048-bit numbers, adjusted based on the specific security requirements.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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