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What is the key size that RSA and Diffie-Hellman are using now that can guarantee secure communication over Internet and will not be able to break by the best available algorithms (NFS & FFS or any others) in feasible time?

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Currently (as of 2017-05-11) 2048-bit keys are most popular for use with RSA, and 2048 bit keys should also be used with classic Diffie-Hellman. These offer about the same security as a symmetric encryption algorithm with 112 bits of security.

Also in common use as of this date are 256 bit Elliptic Curve keys (mostly NIST P-256 and Curve25519) for ECDH/ECDSA/EdDSA which offer the same security as a 128-bit symmetric algorithm.

A good primer on the reasons for these key size differences can be found here.

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A good overview on that matter can always be found on https://keylength.com, which summarises many publications with recommendations for key lengths. Especially NIST SP-800-38 yields good data for your question.

For 112 bit security (which is about the minimum you should use for not extremely important things) you currently (2016) have to use RSA-2048 keys, 2048 bit DH groups or 224 bit elliptic curves.

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    $\begingroup$ This should be the accepted answer as it is more future proof $\endgroup$ – John Demetriou May 12 '17 at 6:11
  • $\begingroup$ I don't know about future-proof, the author seems to be 6 months in the past! (I'm joking, but "currently" is not 2016...) $\endgroup$ – IMSoP May 12 '17 at 10:17
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    $\begingroup$ "Currently" as in "according to the current NIST specification which dates from 2016". Those recommendation do not always get updated every year. $\endgroup$ – mat May 12 '17 at 10:49

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