Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective logic Describing situations in which the
veracity of a statement may be shown to be true without revealing any other information
Etymologies
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Examples

SpiderOak is available for both Linux, Mac and Windows and incorporates 100% zeroknowledge online backup, sync, storage, access and share.

But, that is lost on processliberals who engage in zeroknowledge regulation or, more likely, Hold Harmless! extortion.
Matthew Yglesias » Arctic Seabed Methane Stores Destabilizing

Other applications of combinations of logics include software specification, knowledge representation, architectures for intelligent computing and quantum computing, security protocols and authentication, secure computation and zeroknowledge proof systems, besides its connections to formal ethics and game semantics.

The usual way is to publish the proof, but an alternative way is via zeroknowledge protocols.

There is also the notion of “zeroknowledge” proof.

In general, any mathematical proof can be converted into a zeroknowledge protocol of this form, although it is usually less obvious than the P=NP example

In general, you can always embed your proof into a deduction graph, and then in a zeroknowledge manner convince the world that you know of an appropriate path in the graph.

Which is ultimately irrelevant: if those methods exist, then zeroknowledge protocols, and cryptography in general, are dead.

There is also the notion of “zeroknowledge” proof.

An actual zeroknowledge proof using a rigorous protocol would, presumably, be a reasonable substitute for an actual proof up to a high probability though my knowledge of zeroknowledge proofs are pretty limited.
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