The paper presents efficient algorithms for implementing cryptosystems based on the Tate pairing, focusing on improving pairing evaluation speed and performance. Key contributions include: 1. **Point Tripling**: A new algorithm for point tripling in characteristic 3, reducing the complexity from \(O(m^2)\) to \(O(m)\) steps, leading to faster scalar multiplication. 2. **Square Root Extraction**: An algorithm for computing square roots over \(\mathbb{F}_{p^m}\) in \(O(m^2 \log m)\) steps, which is useful for point compression techniques. 3. **Deterministic Miller's Algorithm**: A deterministic variant of Miller's algorithm for computing the Tate pairing, avoiding irrelevant operations when one argument is restricted to a base field. 4. **Fixed-Base Pairing Precomputation**: Techniques for precomputing coefficients in Miller's algorithm to speed up pairing evaluations, particularly beneficial in characteristic \(p > 3\). The paper also discusses the application of these techniques to MNT curves, which are ordinary (non-supersingular) curves with embedding degree \(k \in \{3, 4, 6\}\). Experimental results show significant improvements in pairing computation times, with BLS signature verification speed improving by about 55 times compared to published timings. The implementations are based on C/C++ and the MIRACL library.The paper presents efficient algorithms for implementing cryptosystems based on the Tate pairing, focusing on improving pairing evaluation speed and performance. Key contributions include: 1. **Point Tripling**: A new algorithm for point tripling in characteristic 3, reducing the complexity from \(O(m^2)\) to \(O(m)\) steps, leading to faster scalar multiplication. 2. **Square Root Extraction**: An algorithm for computing square roots over \(\mathbb{F}_{p^m}\) in \(O(m^2 \log m)\) steps, which is useful for point compression techniques. 3. **Deterministic Miller's Algorithm**: A deterministic variant of Miller's algorithm for computing the Tate pairing, avoiding irrelevant operations when one argument is restricted to a base field. 4. **Fixed-Base Pairing Precomputation**: Techniques for precomputing coefficients in Miller's algorithm to speed up pairing evaluations, particularly beneficial in characteristic \(p > 3\). The paper also discusses the application of these techniques to MNT curves, which are ordinary (non-supersingular) curves with embedding degree \(k \in \{3, 4, 6\}\). Experimental results show significant improvements in pairing computation times, with BLS signature verification speed improving by about 55 times compared to published timings. The implementations are based on C/C++ and the MIRACL library.