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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?

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    $\begingroup$ 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) $\endgroup$
    – SEJPM
    Sep 7, 2021 at 14:30
  • $\begingroup$ 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. $\endgroup$
    – John Smith
    Sep 7, 2021 at 14:38

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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:

$$1≤i≤v:a_i'=(a_i)^{R′_s}$$

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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