Fast primality tests for numbers less than 50 · 109
Mathematics of Computation
Faster primality testing (extended abstract)
EUROCRYPT '89 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
Prime numbers and computer methods for factorization (2nd ed.)
Prime numbers and computer methods for factorization (2nd ed.)
Rabin-Miller primality test: composite numbers which pass it
Mathematics of Computation
Primality testing with fewer random bits
Computational Complexity
The Rabin-Monier theorem for Lucas pseudoprimes
Mathematics of Computation
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Handbook of Applied Cryptography
Handbook of Applied Cryptography
On the difficulty of finding reliable witnesses
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
On Probable Prime Testing and the Computation of Square Roots mod n
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
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Although the Miller-Rabin test is very fast in practice, there exist composite integers n for which this test fails for 1/4 of all bases coprime to n. In 1998 Grantham developed a probable prime test with failure probability of only 1/7710 and asymptotic running time 3 times that of the Miller-Rabin test. For the case that n ≡1 mod 4, by S. Müller a test with failure rate of 1/8190 and comparable running time as for the Grantham test was established. Very recently, with running time always at most 3 Miller-Rabin tests, this was improved to 1/131040, for the other case, n ≡ 3 mod 4. Unfortunately the underlying techniques cannot be generalized to n ≡ 1 mod 4. Also, the main ideas for proving this result do not extend to n ≡ 1 mod 4. Here, we explicitly deal with n ≡ 1 mod 4 and propose a newprobable prime test that is extremely efficient. For the first round, our test has average running time (4 + o(1)) log2 n multiplications or squarings mod n, which is about 4 times as many as for the Miller-Rabin test. But the failure rate is much smaller than 1/44 = 1/256. Indeed, for our test we prove a worst case failure probability less than 1/1048350. Moreover, each iteration of the test runs in time equivalent to only 3 Miller-Rabin tests. But for each iteration, the error is less than 1/131040.