# How do Latin squares relate to quantum cryptography?

About ten years ago I became interested in Latin squares. A friend who was at the Clay Institute at the time mentioned that applications for Latin squares were exploding, and that they were being utilized in quantum cryptography.

I've found a paper discussing use of Latin squares in cryptography in general (p.8 in linked doc), but can't find direct info on their uses in quantum cryptography.

This paper also makes reference to a paper entitled: "A message authentication code based on Latin Squares" from the Australasian Conference on Information Security and Privacy.

However I'm having trouble finding direct references to Latin squares in quantum cryptography.

Basically, I'm trying to get a high-level understanding of this association, if it is indeed accurate. Any info or insight would be appreciated.

There is a conjecture that a full set of mutually orthogonal Latin squares' existence is linked to the existence of MUBs. I am unsure if there is support for this beyond experimental evidence.

Mutually Unbiased Bases (MUBs) introduced by Schwinger are used in quantum measurements and also quantum cryptography. MUBs are $d+1$ Orthonormal bases of $\mathbb{C}^d$ where the inner product of any pair of vectors between two distinct bases is $1/\sqrt{d}$ in magnitude.

For example, constructing a complete set of mutually unbiased bases in the $\mathbb{C}^d$ with $d$ composite and not a prime power has not been achieved yet, not even for $d=6.$

See the PhD thesis of Joanne Hall here

• Thanks for the info and link. I've been reading about MOLS for a while now but MUBs are new to me. The idea that "Complete sets of MOLS and complete sets of MUBs are `similar in spirit'" is enormously helpful. – DukeZhou Jan 5 '17 at 19:54
• attached link to paper gives certificate warning :( – c69 Jul 27 '18 at 5:06
• @c69: does this mean you can't download it? – kodlu Jul 27 '18 at 5:18
• i have full window warning in browser and I'm too scared to click it ) – c69 Jul 27 '18 at 5:20
• I've put it up on this link: filedropper.com/hall – kodlu Jul 27 '18 at 5:26