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Is ECDSA secure from quantum computers? Will quantum computers be able to calculate it's private keys from the knowledge of the public key once there are working quantum computers?

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Short answer: Yes.

Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves.

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    "Yes" is a vague answer because the asker posed two opposite questions: "Is ECDSA breakable by" and "Is ECDSA secure from" quantum computers. – ruief Apr 10 '18 at 0:51
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To quote from pqcrypto.org: "Imagine that it's fifteen years from now. Somebody announces that he's built a large quantum computer. RSA is dead. DSA is dead. Elliptic curves, hyperelliptic curves, class groups, whatever, dead, dead, dead." Worth a visit to their site to understand what crypto looks like after the (hypothetical) success of quantum computing.

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