This paper presents new methods for generating and using "strong trapdoors" in cryptographic lattices, which are simple, efficient, easy to implement, and asymptotically optimal with small hidden constants. The new trapdoor is a matrix that allows efficient inversion of the LWE and SIS functions, and enables secure delegation of trapdoors. The methods improve upon previous techniques in terms of practical performance and output quality. The new trapdoor is smaller and more efficient than traditional bases, and allows for faster and more efficient cryptographic schemes. The paper also shows how these methods can be extended to the ring setting, where functions require only quasi-linear time and space. The new trapdoor generation and inversion algorithms are simple, efficient, and can be used in a wide range of cryptographic applications, including digital signatures and CCA-secure encryption. The paper also provides a detailed comparison of the new methods with previous ones, showing significant improvements in parameters and efficiency. The new methods are based on a new type of trapdoor that is not a basis but a matrix that allows for efficient inversion and sampling. The paper also discusses the theoretical and practical implications of these improvements, including their impact on the security and efficiency of lattice-based cryptography.This paper presents new methods for generating and using "strong trapdoors" in cryptographic lattices, which are simple, efficient, easy to implement, and asymptotically optimal with small hidden constants. The new trapdoor is a matrix that allows efficient inversion of the LWE and SIS functions, and enables secure delegation of trapdoors. The methods improve upon previous techniques in terms of practical performance and output quality. The new trapdoor is smaller and more efficient than traditional bases, and allows for faster and more efficient cryptographic schemes. The paper also shows how these methods can be extended to the ring setting, where functions require only quasi-linear time and space. The new trapdoor generation and inversion algorithms are simple, efficient, and can be used in a wide range of cryptographic applications, including digital signatures and CCA-secure encryption. The paper also provides a detailed comparison of the new methods with previous ones, showing significant improvements in parameters and efficiency. The new methods are based on a new type of trapdoor that is not a basis but a matrix that allows for efficient inversion and sampling. The paper also discusses the theoretical and practical implications of these improvements, including their impact on the security and efficiency of lattice-based cryptography.