# FeRHA: Fuzzy-Extractor-Based RF and Hardware Fingerprinting Two-Factor Authentication
## 3 Preliminaries
### Fuzzy Extractor  
Biometric data are inherently noisy and have limitations in terms of achieving high precision. To overcome these limitations, fuzzy extractors are pursued in security applications to enable stable matching. Fuzzy extractors transform biometric data into random strings, while secure sketches (SS) are used to build fuzzy extractors [50]. The secure sketches generate public helper data strings that correct input errors with minimal information leakage. If the _Hamming distance_ between the _input_ to the fuzzy extractor and the observed one, \(\widehat{input}\), is less than \(T\) during the reproduction process, \(\widehat{input}\) can be corrected, even if there are \(T\) bit errors [51; 52; 53].  
Fuzzy extractors have been widely utilized in hardware-based security primitives, such as PUFs and RF fingerprinting. Specifically, these extractors can rectify the erroneous bits present in the PUF response caused by environmental noise. Additionally, they can also be employed in RF fingerprinting to alleviate the impact of noise during radio signal generation, which can potentially affect the identification of the transmitter (node ID). For instance, in narrow-band systems, the oscillator imperfections of carrier frequency offset (CFO) and phase noise, mixer imperfections of in-phase (I) and quadrature (Q) imbalance, power amplifier (PA) non-linearity, and antenna patterns constitute the RF impairments [54]. These impairments can distort the signal emitted by a device, and a receiver captures the physical waveform to extract these impairments to determine the sender's identity. By utilizing a fuzzy extractor, the noise affecting the physical waveform can be mitigated, thereby enhancing the accuracy in identifying device IDs [55]. Thus, the integration of fuzzy extractors in hardware-based security primitives offers a robust solution to the issues of environmental noise and RF impairments.  
Figure 2 shows the architecture of the adapted fuzzy extractor in this paper, where \(\mathbf{M}\) refers to the original data. As shown in the figure, the generation procedure takes the original template, \(\mathbf{M}\), as input and returns a public helper string, \(S\). The reproduction procedure takes noisy data, \(\mathbf{M}^{\prime}\), and public helper string, \(S\), as input, and outputs \(K\) if \(\mathbf{M}\) and \(\mathbf{M}^{\prime}\) are close, s.t. _Hamming distance_ (\(\mathbf{M}\),\(\mathbf{M}^{\prime}\)) \(\leq T\)[51; 56]. In the context of PUFs, consider a response with 63 bits (\(N=63\)). Assuming a key size of 36 bits (\(K=36\)), the generation process involves feeding the \(K\) bit into the BCH encoder, resulting in a 63-bit value. This value is then subjected to an XOR operation with the original (i.e., non-noisy) 63-bit response. The \((N-K)\) bit of the resulting value, which is 27 bits in this instance, constitutes a helper string, \(S\). In the presence of noisy data, \(\mathbf{M}^{\prime}\), the helper string, \(S\), is XORed with the noisy 63-bit response, \(\mathbf{M}^{\prime}\), as illustrated in Figure 2. The resulting value is then subjected to BCH decoding, yielding the noise-mitigated version of the PUF response, represented as \(\mathbf{M}^{\prime}\).  
When implementing a fuzzy extractor scheme, it is imperative to take into account two critical factors: information reconciliation and privacy amplification. The former guarantees the removal of noise from the collected noisy data, while the latter ensures the uniform distribution of derived key bits. Recent works have utilized a cryptography function, such as SHA-256, and BCH code to satisfy these fundamental requirements [57; 58]. Our proposed protocol employs a fuzzy extractor proposed by [PERSON] et al. [56]. This approach incorporates a Bose-Chaudhuri-Hocquenghem (BCH) in the secure sketch to address the issue of compensating noise in biometrics while ensuring an information-theoretic security of \(K\), where \(K\) refers to the key size [59]. The data perturbation refers to either PUF response bits or RF fingerprinting signals.