Coupling Functions Enable Secure Communications

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Coupling Functions Enable Secure Communications

Tomislav Stankovski, Peter V. E. McClintock, and Aneta Stefanovska

Phys. Rev. X 4, 011026 – Published 26 February 2014

Abstract

Secure encryption is an essential feature of modern communications, but rapid progress in illicit decryption brings a continuing need for new schemes that are harder and harder to break. Inspired by the time-varying nature of the cardiorespiratory interaction, here we introduce a new class of secure communications that is highly resistant to conventional attacks. Unlike all earlier encryption procedures, this cipher makes use of the coupling functions between interacting dynamical systems. It results in an unbounded number of encryption key possibilities, allows the transmission or reception of more than one signal simultaneously, and is robust against external noise. Thus, the information signals are encrypted as the time variations of linearly independent coupling functions. Using predetermined forms of coupling function, we apply Bayesian inference on the receiver side to detect and separate the information signals while simultaneously eliminating the effect of external noise. The scheme is highly modular and is readily extendable to support different communications applications within the same general framework.

Received 16 August 2013

DOI:https://doi.org/10.1103/PhysRevX.4.011026

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Tomislav Stankovski, Peter V. E. McClintock, and Aneta Stefanovska^*

Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

^*Corresponding author. aneta@lancaster.ac.uk

Popular Summary

Secure encryption is an inescapable need of modern life, employed in mobile communication, sensor networks, bank transactions, the Internet, car keys, and so on. In each case, we must be able to pass information securely such that only the intended recipient can decrypt and understand it. Rapid progress in illicit decryption, however, leads to a continuing race for new encryption schemes that are harder and harder to break. In this theoretical paper, we offer a novel encryption scheme, taking inspiration from a seemingly unrelated area of science, biology, that is radically different from any earlier schemes and highly resistant to conventional methods of attack.

The inspiration for our scheme comes from the recent discovery of the time-varying nature of the cardiorespiratory coupling in humans. The idea is for the sender to encrypt information at the transmitter as time variations of the coupling functions between a pair of dynamical systems, e.g., electronic oscillators or lasers, analogous to the heart and lungs. At the receiver end, the information is decrypted with another pair of dynamical systems, identical to those in the transmitter and interacting via the same coupling functions, using Bayesian inference. The use of coupling functions in this way confers an unbounded number of encryption possibilities. We demonstrate that the scheme enables more than one signal to be transmitted/received simultaneously and that it is exceptionally robust against external noise.

The new encryption scheme is quite general, and we expect it to be applicable to many different communications technologies. It allows great freedom in the encryption process without changing the qualitative state of the systems, it is highly modular, and it is readily extendable to support a diversity of different applications within the same conceptual framework.

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Vol. 4, Iss. 1 — January - March 2014

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Figure 1
Schematic of the communications scheme. Messages $s_{1}, \dots, s_{n}$ are encrypted through their modulation of the coupling functions connecting the two oscillators of the transmitter. Only two signals are transmitted through the public domain. For the particular realization of the cipher discussed in the text, Eqs. (5) and (6), they are $x_{1}$ and $y_{2}$ . The receiver consists of the same kind of oscillators and the same coupling functions (effectively forming the privately shared key) as the transmitter and uses time-evolving Bayesian inference to reconstruct $s_{1}, \dots, s_{n}$ .
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Figure 2
The communication procedure based on coupled chaotic oscillators [Eqs. (5) and (6)]. (a) The transmitted binary signal $s_{0} (t)$ acting as a $ρ$ variable in the Lorenz oscillator. Lissajous curves qualitatively indicating the existence of generalized synchronization during (b) synchronized and (c) nonsynchronized intervals. The gray arrows point to synchronization intervals in the $s_{0} (t)$ signal. (d) The largest Lyapunov exponent detected from the driven Lorenz oscillator serves as an index of time-evolving generalized synchronization. (e) Simultaneously transmitted signals $s_{1} (t)$ and $s_{2} (t)$ . For comparison, the original signals $s_{0} (t)$ , $s_{1} (t)$ , and $s_{2} (t)$ are presented in dark color below the received ones shown in light transparent color in (a) and (e)—note that in some intervals they are indistinguishable.
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Figure 3
Example of how noise affects the communication of a binary signal using the model [Eqs. (5) and (6)]. (a) Deviations of the demodulated signal from the initial binary states due to noise, presented as compact box plots (in terms of descriptive statistics: median, quartiles, max, and min) as a function of signal-to-noise ratio (SNR). (b) Bit-error rate (BER) of the coupling-function scheme as a function of SNR. (c) For comparison, BER of the signal-masking scheme as a function of SNR. In (a) and (b) the transmitted signal for coupling-function encryption has the two binary values $s_{1} (t) = {- 2.5, 2.5}$ , while in (c) for signal-masking encryption, $s_{3} (t) = {0, 5}$ . In each run, $1 0^{3}$ randomly ordered binary symbols are sent.
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Figure 4
Transmission of ten pseudorandom binary signals encrypted in different coupling functions. The high values (binary ‘1’) at the transmitter, prior to encrypting, are indicated by grey shading. The received signals, after decrypting, are shown by red lines, each of which (a-j) represents one information signal $s_{i} (t)$ . The particular coupling functions that were used for encrypting each signal are indicated on the ordinate axis. Each bit is transmitted during an interval of 0.2 s, and four bits are transmitted in total for each signal e.g., $s_{7} (t) \to 1010$ . The information could correspond either to ten separate serially-encoded messages, or to a single 10-bit parallel-encoded message, but all of it is being transmitted simultaneously. The bits are indicated by $m_{1} - m_{4}$ on the top of the figure.
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