Computational aspects in the generation of higher-order safe primes

First, an introduction on the current trends of research about special primes is provided. Then, the definition and basic properties of safe primes are presented, extending the concept to higher-order safe primes. An explicit formula to compute the density of this class of primes in the set of the integers is also presented. Finally, explicit conditions are provided permitting the computation of safe primes of arbitrary order.

[1]  J. M. Pollard,et al.  Theorems on factorization and primality testing , 1974, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  R. Horn,et al.  A heuristic asymptotic formula concerning the distribution of prime numbers , 1962 .

[3]  Alfred Menezes,et al.  Handbook of Applied Cryptography , 2018 .

[4]  Dan Boneh,et al.  Cryptanalysis of RSA with private key d less than N0.292 , 1999, IEEE Trans. Inf. Theory.

[5]  Manuel Blum,et al.  A Simple Unpredictable Pseudo-Random Number Generator , 1986, SIAM J. Comput..

[6]  Dan Boneh,et al.  Breaking RSA May Not Be Equivalent to Factoring , 1998, EUROCRYPT.

[7]  Ueli M. MaurerInstitute.  Some Number-theoretic Conjectures and Their Relation to the Generation of Cryptographic Primes 1 , 1992 .

[8]  Michael J. Wiener.  Safe Prime Generation with a Combined Sieve , 2003, IACR Cryptol. ePrint Arch..

[9]  Harvey Dubner,et al.  Large Sophie Germain primes , 1996, Math. Comput..

[10]  Tony Forbes.  Prime clusters and Cunningham chains , 1999, Math. Comput..

[11]  David Naccache.  Double-Speed Safe Prime Generation , 2003, IACR Cryptol. ePrint Arch..

[12]  Dan Boneh,et al.  TWENTY YEARS OF ATTACKS ON THE RSA CRYPTOSYSTEM , 1999 .

[13]  Paul T. Bateman,et al.  waring's problem for algebraic number fields and primes of the form $(p^{r} - 1)/(p^{r} - 1)$ , 1962 .

[14]  Preda Mihailescu,et al.  Fast Generation of Provable Primes Using Search in Arithmetic Progressions , 1994, CRYPTO.

[15]  Jan Camenisch,et al.  Proving in Zero-Knowledge that a Number Is the Product of Two Safe Primes , 1998, EUROCRYPT.

[16]  Paulo Ribenboim,et al.  The little book of big primes , 1991 .

[17]  Ueli Maurer,et al.  Fast Generation of Secure RSA-Moduli with Almost Maximal Diversity , 1990, EUROCRYPT.

[18]  Hugh C. Williams,et al.  A Note on Shanks's Chains of Primes , 2000, ANTS.

[19]  R. Guy.  Unsolved Problems in Number Theory , 1981 .

[20]  J. Littlewood,et al.  Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes , 1923 .

[21]  H. C. Williams,et al.  A $p+1$ method of factoring , 1982 .

[22]  Jean-Sébastien Coron,et al.  Deterministic Polynomial-Time Equivalence of Computing the RSA Secret Key and Factoring , 2006, Journal of Cryptology.

[23]  U. Maurer.  Fast generation of prime numbers and secure public-key cryptographic parameters , 1995, Journal of Cryptology.

[24]  Samuel Yates.  SOPHIE GERMAIN PRIMES , 1991 .