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I know most cryptography Monero uses relies mostly on EC crypto which would be easily broken by a quantum computer powerful enough. What options does Monero have in the future to protect itself?

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Hash functions are not (as) vulnerable to attacks made possible by the existence of quantum computers, as the quickest way to break a hash function is via Grover's algorithm, which provides "only" a quadratic speedup, unlike other quantum algorithms, which can provide exponential speedup over their classical counterparts.

So if you never reveal your public key until you spend all the money from that address, and reveal only the address, you might be okay, as address -> public key would require inverting a hash, which we can assume is still difficult.

Signatures such as edDSA would be broken though, so Monero would probably be best moving to hash-based signature such as Lamport signatures, so that an attacker with a quantum computer can't calculate your private key from your public key/signature and then attempt to double spend your coins.

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  • Diffie-Hellman would also break with quantum computers right? Something like this would be usable with Monero? en.wikipedia.org/wiki/Supersingular_Isogeny_Key_Exchange
    – samwellj
    Commented Dec 3, 2016 at 13:07
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    Did you mean Bitcoin? In Monero, all the one-time public keys are out there in the open so, in theory, if you could crack the one-time spend key, you could spend that output (if it has not been spent already). However, you can't pick a target this way because you can't tell if it has been spent before you crack it. This way, it would likely be a wasted effort.
    – JollyMort
    Commented Dec 3, 2016 at 23:16

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