Timothy Jackson:
> I’ve noted that many (most?) TLS implementations choose their ECDHE curves 
> seemingly without regard to the cipher suite strength. Thus, they'll select 
> an AES256 cipher suite (e.g. TLS_ECDHE_ECDSA_WITH_AES256_SHA384), but then 
> generate an ECDHE key on the P256 curve. This seems odd to me, since the P256 
> curve obviously has a lower security strength than AES256. This seems 
> important issue to resolve because most products (and even TLS libraries) do 
> not allow the administrator to configure the available ECDHE curves, only the 
> cipher suites. Thus, ECDHE may be invisibly undermining the security of your 
> TLS connection.
> 
> Is this an intentional choice by this group for some reason that I haven’t 
> realized yet?
> 
> How would this group feel about a proposal to address this by specifying in 
> the 1.3 specification that implementations must ensure that the strength of 
> the certificate must be >= strength of ECDHE/DHE >= strength of the cipher? 
> Perhaps an equivalency rule for the MAC might also be in order? Apologies if 
> this is already resolved somewhere in the draft RFC. I looked but didn’t find 
> it.
> 
> For what it’s worth, I’ve noticed OpenSSL and other implementations trying to 
> address this by creating a “Suite B Mode”, but that seems to address a 
> specific case but leave the generic case unresolved.
> 
> Cheers,
> 
> Tim
> 
> 
There are good reasons to combine AES256 with ECDHE on the P256 curve.

Please read
https://blog.cr.yp.to/20151120-batchattacks.html

"Bottom line: 128-bit AES keys are not comparable in security to 255-bit
elliptic-curve keys. Is 2^255−19 big enough? Yes. Is 128-bit AES safe?
Unclear."

There is also the paper "Understanding brute force" from Daniel J.
Bernstein.
https://cr.yp.to/papers.html#bruteforce
> 
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