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The issue is that, apart from the Shor’s algorithm, it is hard to find a quantum algorithm, substantially faster than a classical one.

Moreover, since cost of qubits scale exponentially, a slight reduction in the power of a polynomial-time algorithm is not enough to offer a practical benefit.




Ah, walking across glued trees [1]! Not sure of its practical merits however, aha...

[1] - https://arxiv.org/abs/quant-ph/0209131


Cost of qubits does not scale exponentially if we have quantum error correction.


Do you know a proof for that?

I.e., that for n logical qubits, decoherence time goes down polynomially rather than e^(-a n)? AFAIK (and yes, I could be mistaken!) what we deal with is reducing a susbtantially.


Grover's?


“Substantially better” is doing the heavy lifting there. Grover’s gets you a sqrt speedup. Double the size of your hash function (SHA512 instead of SHA256) and you’re protected.


I believe 256 bit symmetric keys and hashes are currently recommended because it's double the classically-secure 128 bit.


Because there is a different sqrt speedup concern regarding the birthday paradox / collision resistance. These combine with Grover's to get you a O(N^(1/3)) speedup when finding a collision, making a 256-bit hash have ~85 bits of security against a quantum adversary.

You'd need to switch to a 384 bit hash function to keep a 128-bit security threshold. In practice, this means SHA-512.




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