## 1 Introduction  
A signature is a behavior-based biometric characteristic that is unique to an individual. It is widely used across various fields, such as finance, administration, and law, as a crucial method for verifying personal identity and conferring legal validity. Since each individual has a unique signature style, it is not a physical characteristic but can still be used to distinguish identity. Since a signature is not a physical characteristic of an individual, it is relatively easier to forge compared to other security elements. Because determining the authenticity of signatures is difficult due to variations in signature styles and the expertise of forgers, many researchers have been attempting to devise signature verification systems. Recent signature verification systems have employed methods such as hidden Markov models [1], support vector machines [2; 3; 4], and neural networks or deep learning [5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. These systems demonstrate good verification performance, but a limitation of these systems is the requirement of multiple signatures from the actual user in training in order to verify the user's signature data. This drawback is particularly pronounced in deep learning-based methods, which require large datasets. To address this issue, data augmentation techniques are being employed, and various methods have been proposed in order to artificially generate signature data and augment training data. [PERSON] et al. (2015) and [PERSON] et al. (2016) proposed signature verification models utilizing transfer learning. They addressed the issue of data scarcity by augmenting the data using traditional image processing techniques, such as rotation, scaling, and flipping. [PERSON] et al. (2017) proposed an online signature augmentation method using sinc interpolation, Gaussian noise addition, signal scaling and rotation, and time warping. [PERSON] et al. (2018) proposed an offline signature augmentation method using linear delta, interpolation-extrapolation, and random noise addition. Numerous studies have also explored the generation of artificial signature data. [PERSON] et al. (2019) introduced a method for creating a signature database using a signature generation algorithm based on spectral analyses. [PERSON] et al. (2020) proposed generating data via duplicating signature features through a synthetic feature generation method based on an artificial immune system rather than augmenting signature images. [PERSON] et al. (2021) developed a method for synthesizing offline signatures, which is inspired by the mortal neuronotor model. Their model is divided into a neuromotor converse model process, which replicates directional planning based on the sigma log-normal model, motor equivalence theory, and kinematic theory. [PERSON] et al. (2021) presented a signature generation model using a generative adversarial network (GAN), demonstrating the creation of signatures that closely resemble original signatures. Although these studies have demonstrated the feasibility of generating signatures that closely resemble original ones, and they have shown that the generation of signature data can positively impact signature verification performance, they largely rely on relatively simple techniques, such as geometric transformations and noise addition. This limits their ability to produce data, such as signatures that truly mimic new signatures created by the same individual. Furthermore, while GAN-based signature synthesis techniques effectively generate new data, the signatures tend to closely resemble training data with respect to the curve shape and stroke length, resulting in a lack of diversity and flexibility. As a result, similarly to earlier methods, the expectation that these techniques can produce signatures that convincingly appear as if they are newly created by the signer is ambitious. In this study, we propose a method for generating signature data using the denoising diffusion probabilistic model (DDPM) ([PERSON] et al., 2014).