20210324, 17:50  #56 
Apr 2020
2^{2}·5^{3} Posts 
RSA100 begins with a 1. And in any case, the size of the number has nothing to do with the number of small primes in the factor base, which is unusually low for RSA100, making it harder to find relations.

20210324, 17:52  #57 
"Ben"
Feb 2007
2×1,789 Posts 
In the data set I collected, all C100's started with a 1 or 2. But the metric I'm measuring is independent of input size as it just looks at the factor base primes < 1000. I gathered essentially the same data on a group of random C110s and C120s.

20210324, 21:02  #58 
Tribal Bullet
Oct 2004
3×1,181 Posts 
Anyone who wants to delve into the standard methods for generating RSA key pairs should read NIST SP80056B, which references FIPS 1864. There are size requirerments on both primes (not just their top bit being set) that force their product to have bit N set in an Nbit modulus.
None of the standards mandate engineering in any resistance to particular subexponential factoring algorithms; that's just a deviation from randomness that could backfire later if those algorithms ever change, and distracts from using key size to provide cryptographic strength. Last fiddled with by jasonp on 20210324 at 21:04 
20210324, 21:14  #59  
"Ben"
Feb 2007
6772_{8} Posts 
Quote:


20210324, 21:18  #60  
"Tilman Neumann"
Jan 2016
Germany
11·43 Posts 
Quote:
Maybe "safe" or "strong" primes may have played a role nonetheless? https://en.wikipedia.org/wiki/Safe_a...s#Cryptography Admittedly, I did not read many posts where RDS explained the selection procedure and can't even find the one I read before. Any references? 

20210324, 22:48  #61  
"Robert Gerbicz"
Oct 2005
Hungary
2·3^{2}·83 Posts 
Quote:
https://www.mersenneforum.org/showpo...7&postcount=33 https://www.mersenneforum.org/showpo...2&postcount=23 

20210325, 14:03  #62  
Tribal Bullet
Oct 2004
3543_{10} Posts 
Quote:
This is a very heavyweight process; for example, if you choose to use pseudoprimes that pass the strong RabinMiller test, the bases to use must be the output of a cryptographic quality random number generator just like P and Q are. There are loads of random bit generators that can be used, and anybody interested in this subject can start with NIST SP80090 

20210325, 21:23  #63  
"Tilman Neumann"
Jan 2016
Germany
111011001_{2} Posts 
Quote:
"We used a hardware RNG with calls to BSAFE to generate the primes" The kind of calls to BSAFE would be interesting, too. https://en.wikipedia.org/wiki/BSAFE That's just what I was talking about, I think... 

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