Abstract: Methods, systems, and computer-readable storage mediums are described for effecting practical use of a dual currency which is a currency that can be electronically created and stored, and further for which there is a secret key (e.g., the secret key of a public key/secret key pair as used in public key cryptography) such that: an entity that does not possess the secret key cannot, in practice, create notes of the currency; and an entity that does possess the secret key can, in practice, create notes of the currency without assistance from entities that do not possess the secret key.

Abstract: Methods, systems, and computer-readable storage mediums are described for effecting practical use of a dual currency which is a currency that can be electronically created and stored, and further for which there is a secret key (e.g., the secret key of a public key/secret key pair as used in public key cryptography) such that: an entity that does not possess the secret key cannot, in practice, create notes of the currency; and an entity that does possess the secret key can, in practice, create notes of the currency without assistance from entities that do not possess the secret key.

Abstract: Computer and communications systems and methods are provided in which a first computing system sends a second computing system a message and an associated deep-string and the second computing system applies a key of a cryptographic system or a one-way function to the deep-string to determine the deep-string's deep-string-depth. The second computing device then uses the determined deep-string-depth in determining subsequent behavior regarding the message. In some environments, a third computing device may generate and provide deep-strings of various deep-string-depths to the first computing device to ensure more favorable behavior of the second computing device.

Abstract: Computer and communications systems and methods are provided in which a first computing system sends a second computing system a message and an associated deep-string and the second computing system applies a key of a cryptographic system or a one-way function to the deep-string to determine the deep-string's deep-string-depth. The second computing device then uses the determined deep-string-depth in determining subsequent behavior regarding the message. In some environments, a third computing device may generate and provide deep-strings of various deep-string-depths to the first computing device to ensure more favorable behavior of the second computing device.

Abstract: Computer and communications systems and methods are provided in which a first computing system sends a second computing system a message and an associated deep-string and the second computing system applies a key of a cryptographic system or a one-way function to the deep-string to determine the deep-string's deep-string-depth. The second computing device then uses the determined deep-string-depth in determining subsequent behavior regarding the message. In some environments, a third computing device may generate and provide deep-strings of various deep-string-depths to the first computing device to ensure more favorable behavior of the second computing device.

Abstract: Computer and communications systems and methods are provided in which a first computing system sends a second computing system a message and an associated deep-string and the second computing system applies a key of a cryptographic system or a one-way function to the deep-string to determine the deep-string's deep-string-depth. The second computing device then uses the determined deep-string-depth in determining subsequent behavior regarding the message. In some environments, a third computing device may generate and provide deep-strings of various deep-string-depths to the first computing device to ensure more favorable behavior of the second computing device.

Abstract: Methods, systems, and computer-readable storage mediums are described for effecting practical use of a dual currency which is a currency that can be electronically created and stored, and further for which there is a secret key (e.g., the secret key of a public key/secret key pair as used in public key cryptography) such that: an entity that does not possess the secret key cannot, in practice, create notes of the currency; and an entity that does possess the secret key can, in practice, create notes of the currency without assistance from entities that do not possess the secret key.

Abstract: A cryptographic communications system and method. The system includes a communications channel coupled to at least one terminal having an encoding device and to at least one terminal having a decoding device. A message-to-be-transferred is enciphered to ciphertext at the encoding terminal by first encoding the message as a number M in a predetermined set, and then raising that number to a first predetermined power (associated with the intended receiver) and finally computing the remainder, or residue, C, when the exponentiated number is divided by the product of two predetermined prime numbers (associated with the intended receiver). The residue C is the ciphertext.