# Security Level Estimate when Cascading/Compositing Ciphers?

Way back when, DES had a theoretical security level 56-bits. 2-key TDEA provides about 80-bits of security, and 3-key TDEA provides about 112-bits of security. Obviously, those security levels are not linear in the cascading of the underlying block cipher.

# First question

How does one estimate the increase in security level when compositing?

# Second question

Is it necessary that the composition is not closed so that the composite does not form a group? I.e, if A(B(C(x))) = y, then there cannot be a A' such that A'(x) = y. (I believe so).

# Third question

How does this apply to Diffie-Hellman over integers? If I run the Diffie-Hellman algorithm twice with 3072 moduli (i.e., 128-bits of security each) and concatenate the two shared secrets, then how many effective bits of security are present? (Intuitively, I think its around 128-bits or 129-bits).

# Fourth question

How does this apply to RSA? If I run the RSA algorithm twice with 3072 moduli (i.e., 128-bits of security each) and transport two shared secrets, then how many effective bits of security are present? (Intuitively, I think its around 128-bits or 129-bits).

Sorry to ask these questions. The back story has to do with elliptic curves and patents, and I'm trying to figure out if its feasible to exchange/agree/transport an AES-256 key in distinct pieces using traditional crypto primitives while maintaining security levels and avoiding elliptic curves.

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You've loaded too many questions together into one. Please pick one question per question. Your first question is too broad; the answer will depend upon whether you're talking about a block cipher or something else. I suggest you just tell us about the real problem you have; or else ask multiple separate questions. –  D.W. Dec 26 '13 at 2:40
You might find this interesting. –  mikeazo Dec 26 '13 at 2:46
Nowadays the reason for cascading is guarding against cryptoanalytic advances against one primitive ensuring that the whole system is as secure as the strongest primitive. It's not a good way to raise the security level. –  CodesInChaos Dec 26 '13 at 10:04

## First question

It depends upon the scheme. It also depends upon what kind of composition you are talking about. Triple-DES uses multiple encryption. That's one kind of composition. The approaches you talk about in your third and fourth questions are a different sort of composition. They will have different security properties.

## Second question

Yes. If the operation forms a group, then using multiple encryption does not add security.

## Third question

This provides only 128-129 bits of security, but not for the reasons you expected. The reason is Diffie-Hellman with a 3072 bit modulus provides only about 128 bits of security; one can break Diffie-Hellman with a 3072 bit modulus using something like $2^{128}$ work (very roughly). So, one can break two Diffie-Hellman exchanges using $2^{129}$ work (very roughly). In other words, this does not add security.

This is not a practical weakness in practice, because in practice 128 bits of security is more than enough; no one is going to be able to perform a computation that requires $2^{128}$ work (not in my lifetime). That said, this is a certificational weakness in the protocol, and no one would consider this to be good protocol design; it is rather crummy protocol design.