The paper introduces new methods for generating and using "strong trapdoors" in cryptographic lattices, which are simultaneously simple, efficient, easy to implement (even in parallel), and asymptotically optimal with very small hidden constants. The methods involve a new kind of trapdoor and include specialized algorithms for inverting LWE, randomly sampling SIS preimages, and securely delegating trapdoors. These improvements address the main bottlenecks in a wide range of cryptographic schemes, leading to better efficiency and security compared to previous methods. The new trapdoor structure can be exposed in applications, further simplifying and improving efficiency. The paper demonstrates the applicability of these methods with new digital signature schemes and CCA-secure encryption schemes, which have better efficiency and security than previously known lattice-based constructions. The main contributions include a new method for trapdoor generation, simpler and more efficient inversion algorithms, tighter parameters, and a new, smaller trapdoor that is at least 4 times smaller in storage than a basis of corresponding quality. The techniques also extend to the ring setting, where functions require only quasi-linear space and time to specify and evaluate.The paper introduces new methods for generating and using "strong trapdoors" in cryptographic lattices, which are simultaneously simple, efficient, easy to implement (even in parallel), and asymptotically optimal with very small hidden constants. The methods involve a new kind of trapdoor and include specialized algorithms for inverting LWE, randomly sampling SIS preimages, and securely delegating trapdoors. These improvements address the main bottlenecks in a wide range of cryptographic schemes, leading to better efficiency and security compared to previous methods. The new trapdoor structure can be exposed in applications, further simplifying and improving efficiency. The paper demonstrates the applicability of these methods with new digital signature schemes and CCA-secure encryption schemes, which have better efficiency and security than previously known lattice-based constructions. The main contributions include a new method for trapdoor generation, simpler and more efficient inversion algorithms, tighter parameters, and a new, smaller trapdoor that is at least 4 times smaller in storage than a basis of corresponding quality. The techniques also extend to the ring setting, where functions require only quasi-linear space and time to specify and evaluate.