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I was thinking about how to use the homeomorphic property of that cryptosystem to achieve not a "semantic security" but a less ambitous obfuscation of the source code (an ipotetical scenario in which a third party execute the obfuscated code, and get a still obfuscated result). So i was asking myself how much computation time to break a small keysize? i am sure there is a way to calculate "resistance" per bit but i don't know how.

Skary
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  • "homeomorphic" is a thing, but only distantly related to homomorphic. There's a wide gap between obfuscating code in a way allowing it's execution, and what Paillier encryption achieves. Thus the goal is not clear. Is it asked the estimated cost of breaking Paillier encryption? – fgrieu May 06 '21 at 10:13
  • @fgrieu yes my question is about that. i just added a bit of context on the why i would use short (and thus weak) keys. To rephrase the question i would ask : how much it cost, to break a 64bit keypair? and a 128bit? and so on (till 256 or so because above then seems to me too costly for what i am thinking about) – Skary May 06 '21 at 10:24

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As far as we know, the most efficient attack against well-implemented Paillier encryption is to factor the public modulus $n$ part of the public key. That's also the best known attack against RSA, thus is well-studied.

For history of factorization open records, see this. In 2020, CADO-NFS has factored a 828-bit $n$ with "roughly 2700 core-years, using Intel Xeon Gold 6130 CPUs as a reference (2.1GHz)".

A smaller data point: CADO-NFS factors a 337 bit $n$ in 5 mn on a single 32-core system.

How much does it cost to break a 64bit keypair? And a 128bit? And so on till 256 or so?

Nothing worth consideration if there is a single such problem to solve.

fgrieu
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  • Thanks for the reference and explanation. So seems that under 512bit it's so simple that does not constitute even a small barier valuable for obfuscation – Skary May 06 '21 at 10:43