Reliable Communication Across Parallel Asynchronous Channels with Glitches

Abstract

Transmission across asynchronous communication channels is subject to laser injection attacks which cause glitches, pulses that are added to the transmitted signal at arbitrary times, and delays. We present self-synchronizing coding schemes with low latency at the receiver that require no acknowledgement and can decode transmissions subject to random delays and distorted by random glitches.

The research of the second author was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 923/16). A preliminary version of part of this work was presented at TRUEDEVICE 2016, Barcelona, Spain.

A Appendices

1.1 A.1 A Sketch of the Proof that the Transmission of Information Is Practically Error Free

We consider a transmitter that makes use of constant-weight codes appropriate to a Z-channel and a decoder that looks for a valid codeword whose ones are contained in a buffer that maintains an ordered list of the pulses seen so far on the channel’s wires and that were not identified as true pulses or glitches during the decoding of previous transmissions. (In our channel, the ones of the actual transmission are always present. Glitches add additional ones.)

When the ones-alone code of Sect. 4.1 is used, all “old” glitches are disregarded. The probability of a glitch affecting any given wire will be less than or equal to $\psi _2$. Using a code appropriate to a Z-channel in which zeros become ones with probability $\psi _2$, one can make certain that each transmission is decoded properly with very high probability.

If one uses the multiple-code based coding framework of Sect. 4.2, then at each stage one must use a code appropriate to a Z-channel with glitch probability $\psi _n$ – where n increases by one after each not-all-zeros transmission. For such a code and large enough N the probability of correctly decoding the transmission can be made to approach one arbitrarily closely. Every $W-1$ transmissions, zeros are transmitted on half the wires, and this reveal the glitches on those wires. We find that the probability of correctly decoding each transmission tends to one as N tends to infinity.

We assume that from time to time we resynchronize so that if there was a mistake in the reception of a transmission, the mistake does not propagate for too long.

1.2 A.2 The Proof that When $R_\mathrm{max}$ Is Maximized, the Density of Ones Tends to $\frac{1}{\phi +2}$

In order to determine the optimal m, we first determine the density of ones among admissible sequences, $\delta$. We then calculate $\log _2(C(N - \lceil \delta \cdot N \rceil , \lceil \delta \cdot N \rceil ))/N$ and show that as $N \rightarrow \infty$ this limit tends to $\log _2(\phi )$.

We know that ${{A}}(W) = {{A}}(w-1) + {{A}}(w-2)$. Let O(W) be the number of ones in all the admissible sequences of length W. Then $O(W) = O(W-1) + O(W-2) + {{A}}(W-2)$ as one builds the sequences of length W by taking all the sequences of length $W-1$ and appending zeros to them and by taking all sequences of length $W-2$ and appending a zero and a one to them. Asymptotically, the number of solutions satisfies ${{A}}(W) \rightarrow {{\alpha }}\phi ^W$. It is easy to see that when calculating the density of ones only this term need be considered. As $\phi$ satisfies the homogeneous version of this recurrence relation, it is not hard to show that

$$\begin{aligned} \frac{1}{\phi + 2}{{\alpha }}W \phi ^W \end{aligned}$$

a particular solution of $O(W) = O(W-1) + O(W-2) + {{\alpha }}\phi ^W$. As the total number of sequences of length W tends to ${{\alpha }}\phi ^W$ and the number of elements in each sequence is W, the average number of ones per element is $\delta = \frac{1}{\phi + 2}$.

Estimating $C(N - m, m)$ while keeping in mind that we intend to take this function’s logarithm and divide by N, we find that

$$\begin{aligned} C(N - m, m) \sim \left( \frac{(N/m) - 1}{(N/m) - 2} \right) ^{N - m} (N/m - 2)^m. \end{aligned}$$

Assuming that as $N \rightarrow \infty$ we choose m to satisfy $m/N \rightarrow \delta$, we find that the rate of the code tends towards

$$\begin{aligned} \left( 1 - \frac{1}{\phi + 2} \right)\times & {} \log _2 \left( \frac{\phi + 1}{\phi } \right) + \frac{1}{\phi + 2} \log _2 \phi \\= & {} \frac{\phi +1}{\phi +2} \log _2(\phi + 1) - \frac{\phi }{\phi + 2} \log _2(\phi ). \end{aligned}$$

Note, however, that $\phi ^2 = \phi + 1$ (as $\phi$ solves the homogeneous recurrence relation), so that the above term is equal to

$$\begin{aligned} 2 \frac{\phi +1}{\phi +2} \log _2(\phi ) - \frac{\phi }{\phi + 2} \log _2(\phi ) = \log _2 \phi . \end{aligned}$$