Forward-secure signatures minimize damage by preventing forgeries for past time periods when a secret key is compromised. Forward-secure signature schemes are useful for various devices such as logging systems, unattended sensors, CCTV, dash camera, etc. Considering sensors equipped with limited resources and embedded real-time systems with timing constraints, it is necessary to design a forward-secure signature scheme with minimal overhead on signer’s side.This paper proposes the first forward secure digital signature schemes with constant complexities in signature generation, key update, the size of keys, and the size of a signature. The proposed algorithms have $$O(k^3)$$O(k3)-time complexity for each signing and key update algorithm and O(k)-size secret keys where k is an RSA security parameter. We prove the security of our proposed schemes under the factoring assumption in the random oracle model and present a concrete implementation of our schemes to demonstrate their practical feasibility.