How are prime numbers used in Crypto?

Could you kindly elaborate on the utilization of prime numbers in the realm of cryptocurrency? I'm particularly interested in understanding how they contribute to the security and functionality of cryptographic systems. Could you also explain how prime numbers are integrated into cryptographic algorithms, and what role they play in ensuring the integrity and privacy of transactions? Additionally, are there any specific examples or case studies that illustrate the effective use of prime numbers in crypto? Thank you for your time and expertise in addressing this topic.