The emergence of metasurfaces provides a secure and efficient platform for optical encryption technology as they have broad prospects in the field of information security. However, the limited number of channels available on metasurfaces and the insufficient security of keys make them vulnerable to attacks by eavesdroppers. In this work, a reprogrammable metasurface optical encryption scheme based on a three-dimensional hyperchaotic system is proposed. The three-dimensional discrete hyperchaotic system has strong ergodicity, initial value sensitivity, and pseudorandomness compared to previous chaotic systems that can pass NIST randomness testing well. Additionally, based on this hyperchaotic property, we designed a metasurface encryption structure based on the geometric phase. The research results show that the introduction of the hyperchaotic system greatly improves the randomness and flexibility of key generation. This scheme can encrypt multiple images with high security. Decryption is only possible when the attacker steals the complete chaotic system and parameters, as well as over 70% of the correct incident light phase information. Our research results have great potential applications in the field of metasurface optical encryption.