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Efficient Generation of Prime Numbers
, 2000
"... The generation of prime numbers underlies the use of most publickey schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality test ..."
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Cited by 12 (4 self)
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The generation of prime numbers underlies the use of most publickey schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation algorithms remain scarcely investigated and most reallife implementations are of rather poor performance. Common generators typically output a nbit prime in heuristic average complexity O(n^4) or O(n^4/log n) and these figures, according to experience, seem impossible to improve significantly: this paper rather shows a simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms. We apply our...
Fast Generation of Prime Numbers of Portable Devices: An Update
 Proceedings of CHES 2006, LNCS 4249
, 2006
"... Abstract. The generation of prime numbers underlies the use of most publickey cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of ..."
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Cited by 3 (1 self)
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Abstract. The generation of prime numbers underlies the use of most publickey cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptography, prime number generation algorithms remain scarcely investigated and most reallife implementations are of dramatically poor performance. We show simple techniques that substantially improve all algorithms previously suggested or extend their capabilities. We derive fast implementations on appropriately equipped portable devices like smartcards embedding a cryptographic coprocessor. This allows onboard generation of RSA keys featuring a very attractive (average) processing time. Our motivation here is to help transferring this task from terminals where this operation usually took place so far, to portable devices themselves in near future for more confidence, security, and compliance with networkscaled distributed protocols such as electronic cash or mobile commerce.
A Simpli ed Quadratic Frobenius Primality Test
, 2005
"... The publication of the quadratic Frobenius primality test [6] has stimulated a lot of research, see e.g. [4, 10, 11]. In this test as well as in the MillerRabin test [13], a composite number may be declared as probably prime. Repeating several tests decreases that error probability. While most of t ..."
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The publication of the quadratic Frobenius primality test [6] has stimulated a lot of research, see e.g. [4, 10, 11]. In this test as well as in the MillerRabin test [13], a composite number may be declared as probably prime. Repeating several tests decreases that error probability. While most of the above research papers focus on minimising the error probability as a function of the number of tests (or, more generally, of the computational e ort) asymptotically, we present a simpli ed variant SQFT of the quadratic Frobenius test. This test is so simple that it can easily be implemented on a smart card. During prime number generation, a large number of composite numbers must be tested before a (probable) prime is found. Therefore we need a fast test, such as the MillerRabin test with a small basis, to rule out most prime candidates quickly before a promising candidate will be tested with a more sophisticated variant of the QFT. Our test SQFT makes optimum use of the information gathered by a previous MillerRabin test. It has run time equivalent to two MillerRabin tests; and it achieves a worstcase error probability of 2 −12t with t tests. Most cryptographic standards require an averagecase error probability of at most 2 −80 or 2 −100, see e.g. [7], when prime numbers are generated in public key systems. Our test SQFT achieves an averagecase error probability of 2 −134 with two test rounds for 500−bit primes. We also present a more sophisticated version SQFT3 of our test that has run time and worstcase error probability comparable to the test EQFTwc presented in [4] in all cases. The test SQFT3 avoids the computation of cubic residuosity symbols, as required in the test EQFTwc. Key Words: smart card, prime number generation, primality testing, quadratic Frobenius test
Close to Uniform Prime Number Generation With Fewer Random Bits
"... Abstract. In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algor ..."
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Abstract. In this paper we analyze a simple method for generating prime numbers with fewer random bits. Assuming the Extended Riemann Hypothesis, we can prove that our method generates primes according to a distribution that can be made arbitrarily close to uniform. This is unlike the PRIMEINC algorithm studied by Brandt and Damg˚aard and its many variants implemented in numerous software packages, which reduce the number of random bits used at the price of a distribution easily distinguished from uniform. Our new method is also no more computationally expensive than the ones in current use, and opens up interesting options for prime number generation in constrained environments. Keywords: Publickey cryptography, prime number generation, RSA, efficient implementations, random bits. 1