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I ask this because it seems more efficient to send change to the original address instead of a new one. Does the cryptography not allow this?

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Can bitcoins be moved from the origin address to the origin address?

Yes. There are no restrictions. (In fact there is no "addresses" in bitcoin protocol)

it seems more efficient to send change to the original address

Why do you think that it can be more efficient? In fact it is not.

Does the cryptography not allow this?

This is not related to cryptography at all.

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