Hardware circuitry defines logic for both Sbox generation and inverse Sbox generation via generating a multiplicative inverse matrix as a truth table for data. The hardware circuitry receives input plain text to be encrypted. The hardware circuitry divides the input plain text to be encrypted. The hardware circuitry feeds multiplicative inverse values generated from the input plain text to a transformer module for performing affine to encrypt the plain text data. The hardware circuitry receives encrypted data to be decrypted. The hardware circuitry divides the encrypted data to be decrypted. The hardware circuitry feeds multiplicative inverse generated from the encrypted data to the transformer module for performing inverse affine to decrypt the encrypted data.

A system for software error-correcting code (ECC) protection or compression of original data using ECC data in a first memory is provided. The system includes a processing core for executing computer instructions and accessing data from a main memory, and a non-volatile storage medium for storing the computer instructions. The software ECC protection or compression includes: a data matrix for holding the original data in the first memory; a check matrix for holding the ECC data in the first memory; an encoding matrix for holding first factors in the main memory, the first factors being for encoding the original data into the ECC data; and a thread for executing on the processing core. The thread includes a Galois Field multiplier for multiplying entries of the data matrix by an entry of the encoding matrix, and a sequencer for ordering operations using the Galois Field multiplier to generate the ECC data.

Operational n-state digital circuits and n-state switching operations with n and integer greater than 2 execute Finite Lab-transformed (FLT) n-state switching functions to process n-state signals provided on at least 2 inputs to generate an n-state signal on an output. The FLT is an enhancement of a computer architecture. Cryptographic apparatus and methods apply circuits that are characterized by FLT-ed addition and/or multiplication over finite field GF(n) or by addition and/or multiplication modulo-n that are modified in accordance with reversible n-state inverters, and are no longer known operations. Cryptographic methods processed on FLT modified machine instructions include encryption/decryption, public key generation, and digital signature methods including Post-Quantum methods. They include modification of isogeny based, NTRU based and McEliece based cryptographic machines.

Embodiments relate to the emulation of the effect of Forward Error Correction (FEC) codes, e.g., GF.sup.10 Reed Solomon (RS) FEC codes, on the bit error ratio (BER) of received Pseudo-Random Binary Sequences (PRBS) patterns. In particular, embodiments group errors into RS-FEC symbols and codewords in order to determine if the errors are correctable. By emulating the error correction capabilities of FEC codes in order to determine which errors are correctable by the code, embodiments afford a more accurate representation of the post-FEC BER of RS FEC codes from links carrying PRBS patterns. This FEC code emulation provides error correction statistics, for stand-alone use or for error correction in connection with Bit Error Rate Testers (BERTs).

Systems and methods are disclosed for processing data. In one exemplary implementation, there is provided a method of generating H output data from W data input streams produced from input data. Moreover, the method may include generating the H discrete output data components via application of the W data inputs to one or more transforming components or processes having specified mathematic operations and/or a generator matrix functionality, wherein the W data inputs are recoverable via a recovery process capable of reproducing the W data inputs from a subset (any W members) of the H output data streams. Further exemplary implementations may comprise a transformation process that includes producing an H-sized intermediary for each of the W inputs, combining the H-sized intermediaries into an H-sized result, and processing the H-sized result into the H output data structures, groups or streams.

A processing acceleration system including at least one gate array that performs finite field arithmetic and at least one controller that sends information to the gate array(s) upon a determination that sending the information, performing the finite field arithmetic by the gate array(s), and sending results of the finite field arithmetic to at least one destination is more efficient than general-purpose computing processor(s) performing the finite field arithmetic and sending the results to the at least one destination. The gate array(s) may include field programmable gate array(s), and the destination(s) may include the general-purpose computing processor(s) or storage devices. The finite field arithmetic may include galois field arithmetic such as modular arithmetic, for example as may be used with respect to erasure coding for storage device(s).

Provided is a BCH decoder in which a folded multiplier is equipped. The BCH decoder may include a key equation solver including a plurality of multipliers. The multiplier includes a plurality of calculation blocks configured to perform a calculation operation. Each of the calculation blocks repeatedly performs a calculation operation of a calculation stage for a plurality of calculation stages, outputs one output value on the basis of at least one input value in each calculation stage, and is connected to at least one another calculation block to transfer an output value of a current calculation stage as an input value of the at least one another calculation block in a next calculation stage.

Low complexity partial parallel architectures for performing a Fourier transform and an inverse Fourier transform over subfields of a finite field are described. For example, circuits to perform the Fourier transforms and the inverse Fourier transform as described herein may have architectures that have simplified multipliers and/or computational units as compared to traditional Fourier transform circuits and traditional inverse Fourier transform circuits that have partial parallel designs. In a particular embodiment, a method includes, in a data storage device including a controller and a non-volatile memory, the controller includes an inverse Fourier transform circuit having a first number of inputs coupled to multipliers, receiving elements of an input vector and providing the elements to the multipliers. The multipliers are configured to perform calculations associated with an inverse Fourier transform operation. The first number is less than a number of inverse Fourier transform results corresponding to the inverse Fourier transform operation.

A syndrome calculation circuit receives input data r(x) including data and a parity bit and having a code length n of (2.sup.m-1) bits at maximum which is represented by a Galois field GF(2.sup.m), and performs syndrome calculation so as to meet

s≡α.sup.i+α.sup.j

z≡(α.sup.i+β).sup.−1+β.sup.−1+(α.sup.j+β).sup.−1+β.sup.1   (A)

thereby calculating syndromes s and z. An error position polynomial coefficient calculation circuit calculates the coefficient of an error position polynomial to obtain s×z by multiplying s and z by one multiplier. After that, 2-bit error data positions i and j are specified. Errors at the error data positions i and j of the input data are corrected.

Techniques are described for joint encoding and decoding of information symbols. In one embodiment, a method for joint encoding includes, in part, obtaining a sequence of information symbols, generating a plurality of cyclic codewords each corresponding to a portion of the sequence of information symbols, jointly encoding the plurality of cyclic codewords to generate at least one combined codeword, and providing the combined codeword to a device. The at least one combined codeword may be generated through Galois Fourier Transform (GFT). In one embodiment, a method for joint decoding includes, in part, obtaining a sequence of encoded symbols, wherein the sequence of encoded symbols is generated through GFT, jointly decoding the sequence of encoded symbols using an iterative soft decision decoding algorithm to generate a decoded sequence, transforming the decoded sequence to generate a plurality of cyclic codewords, and decoding the plurality of cyclic codewords to generate a plurality of decoded information symbols.