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It is well known that it's possible to fool Fermat test with Carmichael numbers. But, is it possible to deliberately fool many-rounded Miller-Rabin test by constructing some special number without using brute forcing strategy? I know that one round of Miller-Rabin distinguish with probability 25% that number is prime, but for many rounds brute force will become infeasible.

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In fact even brute force does not work, unless you know what random numbers the Miller-Rabin test will use to test the numbers, because in case of each possible non-prime number, some Miller-Rabin test input will reveal it is composite.

FIPS 186-4 C.3 contains recommended Miller-Rabin number of rounds to use to test the numbers. Those amounts of Miller-Rabin tests are expected to catch composite numbers with overwhelmingly large probability. The document contains useful information about Miller-Rabin and how test is supposed to (probabilistically) protect from this attack (fooling it with composite numbers).

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  • $\begingroup$ Thanks! I just checked openssl source, and it seems to generate random number for each round, as you said. $\endgroup$ – catpnosis Jan 8 '14 at 7:12

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