How to extend operations from numbers to larger "objects" in cryptographic implementations?

in flag

I know I'm not supposed to roll my own crypto, but everyone starts somewhere! I'm implementing the PSI-CA protocol defined in Fast and Private Computation of Cardinality of Set Intersection and Union (Figure 1, Page 5), and I have it (more-or-less) working. My biggest issue is that I only have it working for int64_t types, and nothing else. Ultimately I'd like to compare strings or even arbitrary objects. So I figure I'll have to have an object expose functionality to serialize/hash itself, but then what? How do I extend this algorithm to multi-byte computations?

For example, if I want to use SHA-512 hashes instead of integers, during certain operations, say $\forall i 1\leq i \leq v : a^{\prime}_i = (a_i)^{R^{\prime}_s}$, how do I translate this from operating over integers (trivial) to bytefields? Do I simply perform this same operation for every byte in the field?

SEJPM avatar
us flag
Is anything stopping you from interpreting and using the serialized / hashed version of an object as an integer? (Which of course requires that serialization / hashing preserves equality)
in flag
I think then it wouldn't be resistant to collisions, right? Like SHA-512 has a large range. The paper says I need two hash functions (modeled as ROM in the paper), so I'm not really sure what hash functions would satisfy these requirements to keep the information cryptographically secure.
ng flag

It is not clear to me (without seeing your code) what your current issue is. Glancing through the paper, I see (for example in fig. 1):

  1. (arbitrary objects) $c_1,\dots, c_v$ and $s_1,\dots, s_v$

  2. hashes of these objects, which are written $hc_i = H(c_i)$ and $hs_i = H(s_i)$ (up to a permutation applied to the $s_i$).

All of the remaining operations are in terms of the $hc_i$ and $hs_i$, which appear to be (from context) in $\mathbb{Z}_p^*$, i.e. the arithmetic is standard (although it is likely that you have to choose $p$ large enough such that the discrete logarithm problem is hard, so of $\gg 1000$ bits. In particular you likely should be using bigints rather than u64's. I haven't read closely enough that I know that they're assuming hardness of a DL-type assumption in $\mathbb{Z}_p^*$ though).

Scanning through the other figures, I see a similar story, namely that all arithmetic is done on hashed elements of $\mathbb{Z}_p^*$, rather than arbitrary bytestrings in $\{0,1\}^*$. I don't see your particular example:


as being an issue. For example, this seems to occur in figure 4, where prior to it I see $a_i = (hs_i)^{R_s'}$. In this figure I don't see the definition $hs_i = H(s_i)$, and I am only skimming, but it is plausible that this interpretation is being assumed, i.e. $hs_i$ is the hash of an arbitrary bytestring $s_i$, and is contained in $\mathbb{Z}_p^*$ (rather than $\{0,1\}^*$).

The general answer to

Do I simply perform this same operation for every byte in the field?

is going to be "no" (unless something specifically says to do this). Typically cryptographic protocols work on well-defined mathematical objects (say bit strings in $\{0,1\}^*$), and "chopping these up" to try to force things to work (when this is not specified as part of the protocol) can easily lead to security issues.


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