Governments are interested in radiation signal encryption in projects relating to international safeguards; however, the available algorithms do not suitably address the challenges presented by the increasing computational capability of various actors, which require recent encryption algorithms to be more robust against attack algorithms. Therefore, an algorithmic approach for performing radiation signal encryption employing the nonlinear capabilities of artificial neural networks with the noise-like properties of chaotic systems is proposed herein. The radiation signal digital envelope consists of the encrypted signal such as may be found through gamma spectroscopy, the secret key for the encryption, and the associated hash value. The presented algorithmic approach demonstrates, in an orderly manner, an integrated method for computing this radiation signal digital envelope. Indispensable constituents of encryption include both the construction of a time series with chaotic characteristics and the incorporation of a hash function generator to satisfy the security requirements of confidentiality, authentication, and nonrepudiation. The methodology is demonstrated via the encryption and subsequent decryption of two frequently occurring radiation signals, namely, gamma spectroscopy signals from 60Co and 137Cs. The results obtained demonstrate the capability of the algorithmic approach to integrate artificial neural networks and chaos dynamics to produce the radiation signal digital envelope (for given security requirements).