There is little point in concerning oneself with smooth p-1, p+1, Φ_i(p) values. Randomly-chosen primes will have a negligible chance of having such properties.
Additionally, the more powerful ECM algorithm can be parametrized to generate about p different elliptic curves E_{a,b}(p), where p is found whenever E_{a,b}(p) has a smooth order. Once again, the chance of finding a pair (a,b) that results in such a smooth order is negligible. This is, in a nutshell, the result of Rivest and Silverman [1] dispelling the need to jump through many hoops to generate strong primes.
Right, so what you're saying is that when the primes are smaller, as they used to be able to be, strong (and safe) primes were a good idea because the earlier methods of factoring could exploit their potential weaknesses. However, it's now the case that with ECM and GNFS methods of factoring we've had to move to much, much larger primes, so the weaknesses are no longer relevant.
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There is little point in concerning oneself with smooth p-1, p+1, Φ_i(p) values. Randomly-chosen primes will have a negligible chance of having such properties.
Additionally, the more powerful ECM algorithm can be parametrized to generate about p different elliptic curves E_{a,b}(p), where p is found whenever E_{a,b}(p) has a smooth order. Once again, the chance of finding a pair (a,b) that results in such a smooth order is negligible. This is, in a nutshell, the result of Rivest and Silverman [1] dispelling the need to jump through many hoops to generate strong primes.
[1] http://people.csail.mit.edu/rivest/RivestSilverman-AreStrong...
Right, so what you're saying is that when the primes are smaller, as they used to be able to be, strong (and safe) primes were a good idea because the earlier methods of factoring could exploit their potential weaknesses. However, it's now the case that with ECM and GNFS methods of factoring we've had to move to much, much larger primes, so the weaknesses are no longer relevant.
Thank you.