Abstract: Disclosed is a multi-purpose secure and anonymous payment system based on a variety of cryptographic confidentiality, authentication, and privacy methods. Users pay anonymously over the Internet using their mobile phones supported by the secure SIM card. The SIM cards do not reveal any personal payment information that is not directly necessary for the transaction to either the merchant or the bank. The system allows configuration of different cryptographic methods or hardware components to allow proper balancing of any specific implementation while maintaining strong security and privacy. It is resilient to connection breakdowns and allows users and merchants to recover from such disruptions without maintaining complex transaction states on the SIM card and without financial losses to any of the parties. The system and protocols can also be configured for electronic cash payments with smart cards or software agents on the Internet or at conventional merchant sale terminals.

Abstract: Techniques are described that facilitate cryptographic operations, such as data encryption, signing and others, encryption using a computerized method for multiplying a scalar by a point. In one implementation, a set of random integers is selected, wherein the set comprises at least one integer selected randomly. A string is configured, based in part on the set of random integers. Output in the form of a product of the scalar and the point is then calculated, based on the string.

Abstract: Techniques are described for generating and validating signatures. In an implementation, a method includes generating a signature by utilizing a plurality of isogenies included on a private key and incorporating the signature and a public key on a product, in which the public key is configured to validate the signature.

Abstract: Techniques are disclosed to provide public-key encryption systems. More particularly, isogenies of Abelian varieties (e.g., elliptic curves in one-dimensional cases) are utilized to provide public-key encryption systems. For example, the isogenies permit the use of multiple curves instead of a single curve to provide more secure encryption. The techniques may be applied to digital signatures and/or identity based encryption (IBE) solutions. Furthermore, the isogenies may be used in other applications such as blind signatures, hierarchical systems, and the like. Additionally, solutions are disclosed for generating the isogenies.