## 1 Introduction  
Detecting tampering with digital data is essential for ensuring the authenticity and integrity of digital content, especially in surveillance and legal applications [1]. Various techniques for tampering detection have been developed in fields involving images [2], videos [3, 4, 5], audio [6], and 3D data streams [7]. We focus on audio tampering detection, a crucial aspect of digital forensics that has numerous applications in the legal and business fields. For instance, it can be used to verify the authenticity of audio evidence in court cases, detect fraudulent activities in financial transactions, and ensure the integrity of audio recordings in business meetings [8]. There are two types of audio tampering detection: passive methods [6] and methods based on information hiding [9, 10, 11, 12, 13]. Passive methods extract the environmental features recorded in the audio signal with its contents, such as the microphone's features [14], background noise [15], or electric network frequency [16, 17]. The information-hiding-based methodsembed secret data in the audio signal, and the embedded data are used to verify the audio signal's integrity. Among these methods, we are developing a tampering detection method based on information hiding. In particular, we aim to develop a tamper detection method to locate the tampered part in the audio signal. Information hiding methods for tamper detection must meet several requirements. Since the integrity of the original data is essential, the hiding algorithm should be reversible [18]. In this paper, we mainly focus on reversible audio watermarking, and we list the interpretations of the keywords of this paper as follows:  
* Stego data: Stego data with hidden information are generated by an information-hiding algorithm. The hidden data may be secret media, a fingerprint, etc. * Hash value: A hash value, a fixed-length unique numerical value, is generated by a particular cryptography algorithm, such as the MD5 and SHA1 algorithms. It can be used as a fingerprint for digital content. * Residual: Residuals are floating numbers expressing predictor coefficients generated by the Linear Predictive Coding (LPC) algorithm. The value is the difference between the original value and the predicted value. * Blind detection: Tampering detection can be processed without the original data. To detect tampering with data with probative importance, the following tasks should be solved. * Reversibility: After applying the proposed embedding and extraction algorithm, the original data can be extracted and re-constructed without data loss. * Imperceptibility: This means that distortion should be controlled to be low enough after embedding the payload for integrity verification. * High capacity: This means the positions reserved for embedding should be adequate to embed digested information for verification, such as hash value, fingerprint, and the necessary information for extraction. The algorithm for the hash value is as long as 128 bits, 160 bits, etc. Reversible watermarking or reversible data hiding methods embed extra information in the cover data to completely recover the original cover data from the stego data. In addition to reversibility, the difference between the original and stego signals must be imperceptible so that the forgers do not perceive the anti-tampering information embedded in the audio signal. Other important aspects are blind detection and embedding capacity. Blind detection means we do not need additional information, such as the original signal, for detecting hidden information. Different embedding methods have different embedding capacities; we must ensure adequate capacity to embed the hash information for the input signal. On the contrary, robustness is not considered for this purpose because we expect the embedded information to be changed with a subtle manipulation such as splicing, encoding, and noise addition [19]. To guarantee the integrity of the original data with probative importance, [PERSON] et al.