[FOM] Provable security and foundations of mathematics

José Manuel Rodríguez Caballero josephcmac at gmail.com
Thu Jun 13 13:02:55 EDT 2019

> Tim wrote (concerning provable security):
>   I don't see any f.o.m. issues of substance at stake here.

In the paper that I cited [The Uneasy Relationship. Between Mathematics
and. Cryptography. Neal Koblitz], the part where I see a connexion between
foundations of mathematics and provable security is the following one:

> For mathematicians who study the provable security literature, as Menezes
> and I did, there are several reasons to be uneasy. Most obviously, a
> provable security theorem applies only to attacks of a specified sort and
> says nothing about clever attacks that might not be included in the
> theorem.

So, my descriptive question:

whether or not provable security can be reduced to foundations
> of mathematics.

can be reformulated in a constructive way as follows:

Could be possible to find a security definition in ZFC which includes all
possible security definitions?

Of course, for this question to be well-defined, it is important to have a
grammar in order to generate all the possible security definitions. This
task belongs to foundations of cryptography, but once the grammar is given,
my question is well-defined in the setting of foundations of mathematics.
Here is a reference to security definitions and their formalization:

Koblitz, Neal, and Alfred Menezes. Another look at security definitions.
Advances in Mathematics of Communications 7.1 (2013): 1-38.

Why this question may be interesting? Well, the security definition which
contains all security definitions reminds me the being satisfying all the
positive properties in Godel ontological argument, i.e., God. It is rather
surprising, at least for me, that the existence of God, according to
Godel's definition, implies the consistency of mathematics, here is the
link to the proof:

A Divine Consistency Proof for Mathematics - A submitted work by Harvey
Friedman showing that if God exists (in the sense of Godel), then
Mathematics, as formalized by the usual ZFC axioms, is consistent

So, maybe there is some relationship between the above-mentioned problem in
provable security and the consistency of mathematics. Something like: if
there is a security definition which implies all the security definitions,
then mathematics is consistent (this is not a conjecture, but just an idea).

Kind Regards,
José M.
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