We often embed items from one space into another space, e.g. embedding plaintext items from $\{0, 1\}^m$ into some ring $R_q/\langle x^N + 1 \rangle$ for RingLWE.

What is the inverse of this action called?

(Feel free to move this question to Math.SE or other related. Though I am specifically interested in the opposite of embed in the context of math and crypto, rather than the general English language.)

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    $\begingroup$ There is no universally agreed dedicated term in the cryptographic community, as far as I'm aware of. One can say project, as kodlu suggests; I would personally use "decode", or perhaps something like "the party embeds the ring element back into $\{0,1\}^m$". $\endgroup$ – Geoffroy Couteau Jun 15 '19 at 22:17

Embed usually implies the target algebraic structure is of larger cardinality (assuming finite). So, I would say project.


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