Abstract: In one embodiment, a method, system, and apparatus are described, the method, system, and apparatus including generating metadata to be associated with each block of a series of blocks, the generating including, except for an initial block, receiving: a first block, including a signed block, and a second block to be signed, retrieving a first value including a square of a random number, R?2, multiplying R?2 by a nonce, r, and setting r·R?2 to be a square of a first random number, denoted R2, for the second block, retrieving a second value from the first block, the second value including K-bit vector, E?, determining a bit string value of the second block, M, computing E=hash(R2?M?E?), and determining a signature, Sig, for the second block by calculating Sig=r Sig? SE-E?. Related methods, systems, and apparatuses are also described.

Abstract: In one embodiment, a method, system, and apparatus are described, the method, system, and apparatus including generating metadata to be associated with each block of a series of blocks, the generating including, except for an initial block, receiving: a first block, including a signed block, and a second block to be signed, retrieving a first value including a square of a random number, R?2, multiplying R?2 by a nonce, r, and setting r·R?2 to be a square of a first random number, denoted R2, for the second block, retrieving a second value from the first block, the second value including K-bit vector, E?, determining a bit string value of the second block, M, computing E=hash(R2?M?E?), and determining a signature, Sig, for the second block by calculating Sig=r Sig? SE-E?. Related methods, systems, and apparatuses are also described.

Abstract: A computer program obfuscating system including a processor to provide a computer program including at least one computer program variable, and add an opaque predicate to the computer program to obfuscate the computer program so that the opaque predicate added to the computer program comprises at least one polynomial including a polynomial P, during execution of the obfuscated computer program, the polynomial P is evaluated yielding at least one result including a first result R1, and during execution of the obfuscated computer program, the opaque predicate is evaluated based on the at least one result R1 such that a decision as to whether or not to perform the first command is dependent upon comparing the first result R1 to at least one value in accordance with a predetermined mathematical relationship. Related apparatus and methods are also included.

Abstract: A computer program obfuscating system including a processor to provide a computer program including at least one computer program variable, and add an opaque predicate to the computer program to obfuscate the computer program so that the opaque predicate added to the computer program comprises at least one polynomial including a polynomial P, during execution of the obfuscated computer program, the polynomial P is evaluated yielding at least one result including a first result R1, and during execution of the obfuscated computer program, the opaque predicate is evaluated based on the at least one result R1 such that a decision as to whether or not to perform the first command is dependent upon comparing the first result R1 to at least one value in accordance with a predetermined mathematical relationship. Related apparatus and methods are also included.