Visual cryptography is a cryptographic scheme that allows the decoding of concealed images without any cryptographic computations. The scheme is perfectly secure and easy to implement. It extends to a visual variant of the k out of n secret sharing problem, where any k of n users can reconstruct the image by stacking their transparencies, while fewer than k users gain no information. The basic model involves a printed page of ciphertext and a printed transparency. When the transparency is placed over the ciphertext, the original message is revealed. This is similar to a one-time pad, where each page is decrypted with a different transparency. The system is simple and can be used by anyone without cryptographic knowledge. The visual secret sharing problem can be extended to a k out of n scheme. For example, a 2 out of n scheme can be solved using specific matrices. The 2 out of 2 scheme is the original visual cryptography problem, which can be solved with two subpixels per pixel. However, this can distort the aspect ratio, so four subpixels arranged in a 2x2 array are often used instead. For the 3 out of 3 scheme, specific matrices are used to ensure that any three transparencies stacked together reveal the secret, while fewer do not. This scheme can be generalized to a 3 out of n scheme using specific matrices. The paper presents two general constructions for k out of k schemes. The first uses 2^k subpixels and the second uses 2^{k-1} subpixels. The second construction is proven optimal, requiring at least 2^{k-1} subpixels. For a general k out of n scheme, the paper describes a method to extend a k out of k scheme to a k out of n scheme. This involves using a collection of functions to map subsets of n elements to subsets of k elements, ensuring that any k transparencies reveal the secret while fewer do not. The paper also discusses extensions of the basic model, including visual encryption of continuous tone images and concealing the existence of a secret message. These extensions use specific subpixel arrangements to achieve the desired visual effects. The paper concludes with a discussion of the theoretical bounds on the parameters of the schemes, showing that the relative difference α must be exponentially small as a function of k. The constructions presented provide efficient and secure visual secret sharing schemes for various values of k and n.Visual cryptography is a cryptographic scheme that allows the decoding of concealed images without any cryptographic computations. The scheme is perfectly secure and easy to implement. It extends to a visual variant of the k out of n secret sharing problem, where any k of n users can reconstruct the image by stacking their transparencies, while fewer than k users gain no information. The basic model involves a printed page of ciphertext and a printed transparency. When the transparency is placed over the ciphertext, the original message is revealed. This is similar to a one-time pad, where each page is decrypted with a different transparency. The system is simple and can be used by anyone without cryptographic knowledge. The visual secret sharing problem can be extended to a k out of n scheme. For example, a 2 out of n scheme can be solved using specific matrices. The 2 out of 2 scheme is the original visual cryptography problem, which can be solved with two subpixels per pixel. However, this can distort the aspect ratio, so four subpixels arranged in a 2x2 array are often used instead. For the 3 out of 3 scheme, specific matrices are used to ensure that any three transparencies stacked together reveal the secret, while fewer do not. This scheme can be generalized to a 3 out of n scheme using specific matrices. The paper presents two general constructions for k out of k schemes. The first uses 2^k subpixels and the second uses 2^{k-1} subpixels. The second construction is proven optimal, requiring at least 2^{k-1} subpixels. For a general k out of n scheme, the paper describes a method to extend a k out of k scheme to a k out of n scheme. This involves using a collection of functions to map subsets of n elements to subsets of k elements, ensuring that any k transparencies reveal the secret while fewer do not. The paper also discusses extensions of the basic model, including visual encryption of continuous tone images and concealing the existence of a secret message. These extensions use specific subpixel arrangements to achieve the desired visual effects. The paper concludes with a discussion of the theoretical bounds on the parameters of the schemes, showing that the relative difference α must be exponentially small as a function of k. The constructions presented provide efficient and secure visual secret sharing schemes for various values of k and n.