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Imagine a situation where there are many high-value public keys around, using the same Elliptic Curve group, say $k$ in the millions public keys¹. Can an adversary reasonably find one of the matching private key at much lower cost that finding the private key for a particular one?

What's the best feasible² method? What's it's cost relative to the best known feasible method for one key (that is, I believe, distributed Polard's rho with distinguished points), as a function of $k$ and perhaps the Elliptic Curve group order $n$?


¹ Imagine Bitcoin with secp224k1, and the corresponding ponzi had similar market value.

² Assuming known existing technologies, including supercomputers, GPUs, FPGAs, ASICs, but not quantum computers usable for cryptanalysis.

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    $\begingroup$ I know Kuhn and Struik proved in 2001 (section 4) that Pollard's rho method can compute $k$ discrete logs in $\sqrt k$ time. The first takes the full expected time, the second less, the next even less, etc. $\endgroup$ Jun 25 '21 at 14:31
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    $\begingroup$ The issue with that is that it's from 2001. I expect there's new research, or that new attacks might be cheaper. So I don't really want to answer it with that alone, since it's a 20-year old paper. I just remembered that it's cited in the "Batch discrete logarithms" section of Bernstein's Curve25519 paper and looked it up. Certainly it's an upper-bound on the difficulty though. $\endgroup$ Jun 25 '21 at 14:36
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    $\begingroup$ This is basically the same as this answer $\endgroup$ Jun 25 '21 at 16:04
  • $\begingroup$ @Samuel Neves: thanks for pointing that. Not quite the same maybe: precomputation is not the same as multi-target, because the target(s) are not known when a precomputation starts. In RSA at least, that makes a significant difference: I know no precomputation attack to factor RSA moduli, but there are some (borderline useful) multi-target attacks, like Pollard's p-1. $\endgroup$
    – fgrieu
    Jun 25 '21 at 16:06
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    $\begingroup$ No; finding all $k$ private keys costs $O(\sqrt{kn})$, that is, you save a $\sqrt{k}$ factor compared to solving each log separately. This has been explicitly proved by Yun, but was already the cost of the best attack since 1997 or so (Silverman). $\endgroup$ Jun 25 '21 at 16:16
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Can an adversary reasonably find one of the matching private key at much lower cost that finding the private key for a particular one?

No, and that's provable (and is independent of the technology employed)

Suppose that we had a black box that could take $k$ different public keys $a_1G, a_2G, ..., a_kG$, and recover $a_iG$ (for some $i$) in $o(\sqrt{n})$ time.

Then, here is how we could use that black box to, given one public key $aG$, recover the private key $a$ in $o(\sqrt{n})$ time. We would:

  • Select $k$ random values $r_1, r_2, ..., r_k$, and compute the sequence $r_1(aG), r_2(aG), ..., r_k(aG)$, which (by defining $b_i = r_i a$) can be viewed as $b_1G, b_2G, ..., b_kG$

  • Give the sequence $b_1G, b_2G, ..., b_kG$, which will recover $b_i$

  • We compute $a = r_i^{-1}b_i$, and thus recover the key.

The steps in addition to the invocation of the black box takes $O(k)$ time, which can be ignored for reasonablely sized $k$.

Note that the sequence $b_1G, b_2G, ..., b_kG$ is uniformly distributed, and hence even if the black box is probabilistic, it'll still allow us to recover the public key.

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  • $\begingroup$ That's a variation of something you already told me, and spot on! $\endgroup$
    – fgrieu
    Jun 25 '21 at 21:10

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