# Effects of Kerr Nonlinearity in Physical Unlonable Functions
## 1 Introduction  
Physical unclonable functions (PUFs) based on optical multiple scattering media have attracted considerable attention over the last decades, mainly because they are considered to pave the way for the development of novel more robust cryptographic protocols [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. The optical multiple scattering medium essentially serves as a token, and the internal disorder renders its cloning a formidable technological challenge. At the same time, the response of the token to optical challenges is a complex pattern (speckle), which depends strongly on the internal disorder of the token as well as on the challenge. Different challenges pertain to different parameters of the incoming light, e.g., intensity, point and angle of incidence, wavelength or combinations thereof [1; 5; 9; 10; 12]. Formally speaking, the optical response of the token involves a large number of optical modes, and the speckle is essentially the result of interference of many paths that lead to a particular mode at the output [17]. In the framework of cryptographic applications, the speckle can be used raw as a fingerprint [5; 18], or it can be processed classically to yield a random numerical key [1; 2; 3; 10]. In any case, the sensitivity of the speckle to the potential cloning of the token essentially determines the resistance of the cryptographic protocol under consideration against related attacks. A useful protocol should allow one to distinguish between a token and its clones by comparing their responses to the same challenge (raw speckles or numerical keys).  
All of the aforementioned studies rely on the use of a linear multiple scattering medium as a token, which results in a linear relationship between the electric field at a particular point at the output plane and the electric field at the input plane. This linear relationship may render certain cryptographic protocols vulnerable to cloning attacks, provided the attacker has access to the raw speckle images that are produced by the setup for the particular token. It has been conjectured [1; 18] that such attacks can be thwarted by using nonlinear multiple scattering tokens, thereby breaking the linear dependence of the speckle on the electric field at the input. To the best of our knowledge, this conjecture remains an open question in the field up to date.  
In this work, our aim is to address for the first time this open question by investigating whether and under what conditions multiple scattering media with Kerr nonlinearity can offer some advantage over their linear counterparts with respect to their robustness against cloning. To this end, we focus on the speckle that is produced by a reference token, and we study its correlation to the speckles of different random clone tokens. Given the high sensitivity of the speckle to the internal disorder, we adopt a genericmodel for the (non)linear token, which allows us to reach definite general conclusions through the statistical analysis of the data. Our results suggest that in general, tokens with focusing nonlinearity can be more robust to cloning than linear tokens or tokens with defocusing nonlinearity.