# Chaos-Based Physical Unclonable Functions
## 1 Introduction  
Modern cryptography is facing progressively more attacks directed not on cryptographic algorithms, but on their implementations--even the secured ones [1]. Among many kinds of side-channel attacks (SCAs), there are various ways of retrieving information from memories, where the cryptographic keys are kept [2, 3], and therefore there is a struggle for securing the memories against SCAs [4]. At the same time, there is an increasing demand for secure cryptography with an application in small and inexpensive circuits (like Internet of Things devices, wearables, implantable medical devices, etc. [5, 6]). These are the reasons why PUFs are drawing more and more attention in modern secure electronics--they can make it possible to create \"a vault\" for cryptographic keys, without building an actual vault [7, 8]. The advantages of particular PUFs (as well the chaotic PUF presented in this paper) in the fields of cryptography and security include the following:  
* PUF keys are usually not present in the system (cryptographic keys are not kept in any volatile memory, non-volatile memory, latches nor registers);
* a key is temporarily activated (re-generated) when it is required in the system;
* a key can be activated only by its owner (by the use of owner's initiation vector--the _PUF challenge_);keys are unique and different for every instance of a similar device (programmed in the same way, with the use of the same code, and operating on the same data);
* keys cannot be copied, cloned nor extracted from the device, as well as they are tamper-proof (any attempt of tampering should destroy the keys). Such features can be utilized in various attractive ways:
* unambiguous and incontestable identification of a unit;
* authentication, digital signature, encryption/decryption;
* owner/manufacturer authentication (e.g., for the use of certified updates, preventing hacking);
* immunity to spoofing, cloning, reverse engineering, and man-in-the-middle attacks. The advantages and applications of PUFs initiated a search for methods of harvesting of PUF keys from physical processes. Among proposed solutions, we can find: ring oscillators [7; 9], transient effect ring oscillators [10], dynamic ring oscillators [11], ordering-based ring oscillator [12], convergence time of bistable rings [8], sneak paths in the resistive X-point array [13], power consumption differences of Advanced Encryption Standard Sbox inversion functions [14], occurrence of metastability [15], static memory [16; 17], dynamic memory [18; 19], switching behavior of emerging magneto-resistive memory devices [20], switching of resistive random access memory [21], reduction-oxidation resistive switching memories [22], decay-based Dynamic Random Access Memory [23], locally enhanced defectivity [24], combination of multiplexers and arbiters [25], wireless sensors [26], Complementary Metal-Oxide Semiconductor image sensors [27], nonlinearities of data converters [28], mismatch of capacitor ratios [29], primitive shifting permutation network (barrel shifter) [30], cellular neural networks [31], customized dynamic two-stage comparator [32], and many others. Among the ideas, there are various ways of using ring oscillators for the sake of PUFs (containing an odd number of inverters); however, a ring consisting of an even number of inverters can stabilize in only one of two states when powered up or, more generally, when it is initiated from an unstable state. Such an architecture is called bistable ring PUF (BR-PUF) [33; 34; 35]; however, in this particular application, the inverters were replaced with more suitable cells that provide an easy cell reset (by the use of NOR gates or a dedicated architecture) as well as the ability to chose one of two gates (for the sake of PUF challenges).