Bingham read the Amendment as an absolute bar on quartering in peacetime but not in wartime. I don’t think there is any case law on this, though a look at the original understanding of the Amendment (and the complaints about the practice in the Declaration of Independence) might be helpful.

Well, whenever you do a post about an exotic or dormant constitutional provision, that always should be qualified a little bit. Though if somebody has to be known as “The Third Amendment Guy,” I guess it might as well be me.

There’s another textual ambiguity, though: does the clause “but in a manner to be prescribed by law” apply only “in time of war,” or does it apply to peacetime quartering, too? I.e., is “No Soldier shall, in time of peace be quartered in any house, without the consent of the Owner, … but in a manner to be prescribed by law” a legitimate reading? Or is consent in peacetime an absolute requirement? Douglas, J. seems to have read it as the latter in Griswold (“The Third Amendment, in its prohibition against the quartering of soldiers ‘in any house’ in time of peace without the consent of the owner, is another facet of that privacy”), but this is dicta; and none of the concurring opinions so much as mentions the Third Amendment. Is there other precedent that supports Douglas, J.’s reading, or subsequent case law that’s solidified it?

The assertion was that use of escrowed encryption would never be made mandatory; many people disbelieved this, because there were few (if any) other incentives to adopt it. There was speculation on some cryptographic mailing lists that this might run afoul of the 3rd Amendment — being forced to deploy an aid to wiretapping of your own communications, to aid in surveillance — but I never saw any serious legal analysis, as opposed to amateur theorizing by (generally radical libertarian) cryptographers.

In any event, your point — that Congress can perhaps do this, but not the executive branch — bears on that issue. The cryptography export rules are a creation of two regulatory regimes, the International Trafficking in Arms rules and the Export Administration Rules; while these are (I assume) based on statutes, to my knowledge the definition of covered cryptographic schemes (and there are still such, despite the drastic loosening of the rules around 2000) are regulatory creations, not statutory.