How can the properties of prime numbers be applied to developing new cryptographic algorithms?

Prime numbers have long been a cornerstone in the field of cryptography, particularly in algorithms like RSA which rely on the complexity of factoring large composites. However, as computational power increases and quantum computing looms on the horizon, there is a need to explore innovative uses of prime numbers to ensure secure cryptographic systems. This question invites a discussion on novel approaches to using prime numbers in algorithm design, such as leveraging lesser-known mathematical properties or integrating them with emerging technologies, to develop enhanced security protocols that can withstand future challenges.