Fast and Simple One-Way High-Dimensional Quantum Key Distribution

Kfir Sulimany Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Rom Dudkiewicz School of Computer Science & Engineering, The Hebrew University of Jerusalem, Jerusalem, 91904 Israel Simcha Korenblit Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Hagai S. Eisenberg Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Yaron Bromberg Michael Ben-Or

1 Effective attack against COW

Eve has an effective attack on larger blocks, that can help her extort most of the information given the standard variations of COW. We’ll only consider states where Bob receives at most one signal per block. Eve’s states after interaction will be in 4 groups:

$|V\rangle$ meaning she passed void in all time slots,

$|P_{0,i}\rangle$ meaning she passed $|1\rangle$ in time slot i when $|0\rangle$ was passed by Alice in that time slot,

$|P_{\mu,i-j}\rangle$ meaning Alice sent $|\mu\rangle$ in all time slots i-j, and $|0\rangle$ in the adjacent time slots, and Eve passed $|1\rangle$ in on of the time slots i-j,

$|P_{\mu,edge}\rangle$ meaning Eve passed $|1\rangle$ in a time slot that was part of a series of $|\mu\rangle$ sent by Alice reaching one (or both) of the block’s edges. For example:

\begin{gathered}|\sqrt{\mu},0,0,\sqrt{\mu},\sqrt{\mu},0,\sqrt{\mu},\sqrt{\mu}\rangle_{A}|\varepsilon\rangle_{E}\to\\ |0,0,0,0,0,0,0,0\rangle_{B}|V\rangle_{E}\\ +\sqrt{(1-Q)\mu t}|1,0,0,0,0,0,0,0\rangle_{B}|P_{\mu,edge}\rangle_{E}\\ +\sqrt{Q\mu t}|0,1,0,0,0,0,0,0\rangle_{B}|P_{0,2}\rangle_{E}\\ +\sqrt{Q\mu t}|0,0,1,0,0,0,0,0\rangle_{B}|P_{0,3}\rangle_{E}\\ +\sqrt{(1-Q)\mu t}|0,0,0,1,0,0,0,0\rangle_{B}|P_{\mu,4-5}\rangle_{E}\\ +\sqrt{(1-Q)\mu t}|0,0,0,0,1,0,0,0\rangle_{B}|P_{\mu,4-5}\rangle_{E}\\ +\sqrt{Q\mu t}|0,0,0,0,0,1,0,0\rangle_{B}|P_{0,6}\rangle_{E}\\ +\sqrt{(1-Q)\mu t}|0,0,0,0,0,0,1,0\rangle_{B}|P_{\mu,edge}\rangle_{E}\\ +\sqrt{(1-Q)\mu t}|0,0,0,0,0,0,0,1\rangle_{B}|P_{\mu,edge}\rangle_{E}\end{gathered}

(1)

We treat $\mu<<1$ so when we measure visibility, the case of no photon will be controlled by $V$ , so if we for example measure visibility between time slots 4 and 5, the states on the detectors will be $|P_{\mu,4-5}\rangle\pm|P_{\mu,4-5}\rangle$ and the visibility will be 1. So the only constraint from visibility is that $|P_{\mu,edge}\rangle$ and $|0\rangle$ will be sufficiently closed.

for large enough blocks, the chance for $|P_{\mu,edge}\rangle$ are very small, and Eve can almost always reconstruct the entire qudit