Barrett modular reduction and multiplication are essential primitives for efficient modular computation in cryptographic schemes, including post-quantum standards such as machine learning (ML) key encapsulation mechanism (KEM) and ML-digital signature algorithm (DSA). To protect against faults that compromise correctness and security, we introduce the first efficient fault-detection mechanisms tailored to these operations. For modular reduction, we leverage word-based representations with compact word-sum checks that exploit algebraic input-output relations to ensure computational integrity. For modular multiplication, we adopt a tunable hybrid strategy: early stages apply word-sum checks, while later stages use partial recomputation with encoded inputs, providing robust protection against injected faults. Formal analysis, fault-injection simulations, and hardware/software implementations show that our methods detect a wide range of faults with minimal performance and area overhead. Evaluation results demonstrate overheads of 3.43% and 7.15% for 512-bit and 1024-bit inputs in modular reduction, and 26.47% and 27.22% for 2048-bit inputs in modular multiplication in the number of clock cycles in software. Moreover, in hardware, we observed reasonable overheads: less than 27.5% in area and 2.1% in delay for modular reduction, and less than 23.5% in area and 16.2% in delay for modular multiplication. These results confirm the practicality of our methods for secure yet efficient integration.

After the acceptance of [1], an error was introduced, which we aim to resolve here. The abbreviation ML stands for module lattice-based, not “machine learning.” The first sentence of the first paragraph is corrected from the version that was published in Early Access. It should have read, “Barrett modular reduction and multiplication are essential primitives for efficient modular computation in cryptographic schemes, including postquantum standards such as module lattice-based (ML) key encapsulation mechanism (KEM) and ML-digital signature algorithm (DSA).” In the Introduction, the same correction has been made for the abbreviation ML.