A symmetric cryptography for encrypting and decrypting information is provided, that can be implemented efficiently in hardware or in software. The symmetric cryptography uses a key generator, so that the cryptography is not dependent on a single, static cryptography key. The key generator is a value or collection of values from which the key is generated. The key generator substantially increases the computational complexity of differential cryptanalysis and other cryptographic attacks. In an embodiment, the key generator is updated with one-way functions exhibiting the avalanche effect, which generates an unpredictable sequence of keys used during the encryption or decryption process. In an embodiment, a dynamic key is derived from a key generator with a one-way hash function. In an embodiment, a block cipher uses a different dynamic key to encrypt each block of plaintext, where each key is derived from a different key generator.

A method and system is provided for solving for electromagnetic fields by approximating an electromagnetic function as a sum of basis functions multiplied by coefficients to be determined. The set of equations used to determine the coefficients results from taking derivatives of the action integral with respect to the coefficients (and/or other parameters) and setting the derivative equal to zero, thereby extremizing the action integral.

Methods and systems are provided for performing a secure transaction. Users register biometric and/or other identifying information. A registration code and an encryption key are generated from the biometric information and/or information obtained from a unpredictable physical process and are stored in a secure area of a device and also transmitted to a service provider. A transaction passcode generator may be computed based on the stored registration code. In at least one embodiment, a unique transaction passcode depends upon the transaction information, so that on the next step of that transaction, only that unique transaction passcode will be valid. In an embodiment, the passcode includes the transaction information. In at least one embodiment, if the transaction information has been altered relative to the transaction information stored in the device's secure area, then the transaction passcode sent during this step will be invalid and transaction may be aborted.

Based upon the principles of Turing incomputability, connectedness and novel properties of the Active Element Machine, a malware-resistant computing machine is constructed. Using randomness, the active element machine can deterministically execute a universal Turing machine (universal digital computer program) with active element firing patterns that are Turing incomputable. In some embodiments, if the state and tape (or other memory) contents of the universal Turing machine and the random bits generated from the quantum source are all kept perfectly secret and no information is leaked about the dynamic connections between the active elements, then it is Turing incomputable to construct a translator Turing machine (translator digital computer program) that maps the random firing interpretations back to the sequence of instructions executed by the universal Turing machine. A more powerful computational procedure is created than Turing's computational procedure (digital computer procedure).

An ablation apparatus places electrodes at the perimeter of a cavity. In an embodiment, the alternating electric field is used to expose the cavity to enough energy to ablate the cavity. In an embodiment, two modes are used to expose different regions of the cavity to different amount of power for so that the thermal effect is more uniform. In an embodiment, the electrodes have a relatively large surface area so as to avoid charring the cavity, but are shaped so as to fit within a body orifice. For example, the diameter of the sheathed housing the electrodes during penetration may be only 5.5 mm.