A processing device for federated learning, including: a modular exponentiation module including at least one modular exponentiation engine; a pre-processing module for providing operations corresponding to a plurality of operator modes; a montgomerization module for providing montgomerization operations; a confusion calculation module for providing modular multiplication operations in montgomery space; a montgomery reduction module for providing montgomery reduction operations; and a controller for determining, according to an input operator mode, whether to enable at least two modules out of the pre-processing module, the montgomerization module, the confusion calculation module, and the montgomery reduction module, so as for cooperatively performing the input operator mode together with the modular exponentiation module.

A Montgomery multiplication apparatus (MMA), for multiplying two multiplicands modulo a predefined number, includes a pre-compute circuit and a Montgomery multiplication circuit. The pre-compute circuit is configured to compute a Montgomery pre-compute value by performing a series of iterations. In a given iteration, the pre-compute circuit is configured to modify one or more intermediate values by performing bit-wise operations on the intermediate values calculated in a preceding iteration. The Montgomery multiplication circuit is configured to multiply the two multiplicands, modulo the predefined number, by performing a plurality of Montgomery reduction operations using the Montgomery pre-compute value computed by the pre-compute circuit.

An Integrated Montgomery Calculation Engine (IMCE), for multiplying two multiplicands modulo a predefined number, includes a Carry Save Adder (CSA) circuit and control circuitry. The CSA circuit has multiple inputs, and has outputs including a sum output and a carry output. The control circuitry is coupled to the inputs and the outputs of the CSA circuit and is configured to operate the CSA circuit in at least (i) a first setting that calculates a Montgomery precompute value and (ii) a second setting that calculates a Montgomery multiplication of the two multiplicands.

Systems and methods for end-to-end encryption and compression are described herein. A query is encrypted at a client using a homomorphic encryption scheme. The encrypted query is sent to a server where the encrypted query is evaluated over target data to generate encrypted response without decrypting the encrypted query. The result elements of the encrypted response are grouped, co-located, and compressed, without decrypting the encrypted query or the encrypted response. The compressed encrypted response is sent to the client where it is decrypted and decompressed to obtain the results of the query without revealing the query or results to the owner of the target data, an observer, or an attacker.

Aspects of the present disclosure involve a method, a system and a computer readable memory to generate and use prime numbers in cryptographic operations by determining one or more polynomial functions that have no roots modulo each of a predefined set of prime numbers, selecting one or more input numbers, generating a candidate number by applying one or more instances of the one or more polynomial functions to the one or more input numbers, determining that the candidate number is a prime number, and using the determined prime number to decrypt an input into the cryptographic operation.

Aspects of the present disclosure describe a method and a system to support execution of the method to perform a cryptographic operation involving identifying an N-word number, X=XN−1 . . . X.sub.1X.sub.o, to be squared, performing a first loop comprising M first loop iterations, wherein M is a largest integer not exceeding (N+1)/2, each of the M first loop iterations comprising a second loop that comprises a plurality of second loop iterations, wherein an iteration m of the second loop that is within an iteration j of the first loop comprises computing a product X.sub.a*X.sub.b of a word X.sub.a and a word X.sub.b, wherein a+b=2j+m, j≥0 and m≥0, and wherein all second loops have an equal number of second loop iterations.

Integrated circuits for modular multiplication of two integers for a cryptographic method, and methods for the cryptographic processing of data based on modular multiplication are herein disclosed. For example, an integrated circuit for modular multiplication of two integers for a cryptographic method has a processor that represents the integers to be multiplied in Montgomery representation with a specified Montgomery representation parameter and a specified modulus, and calculates the result of the modular multiplication of the integers to be multiplied in Montgomery representation iteratively from the least significant word to the most significant word, where for a word calculated in an iteration, the product of the word with a specified factor is added to the words of subsequent iterations, the specified factor given by the product of the negative inverse of the least significant word of the modulus and the modulus, without the least significant word of the product, plus one.

Methods and apparatus for optimization techniques for modular multiplication algorithms. The optimization techniques may be applied to variants of modular multiplication algorithms, including variants of Montgomery multiplication algorithms and Barrett multiplication algorithms. The optimization techniques reduce the number of serial steps in Montgomery reduction and Barrett reduction. Modular multiplication operations involving products of integer inputs A and B may be performed in parallel to obtain a value C that is reduced to a residual RES. Modular multiplication and modular reduction operations may be performed in parallel. The number of serial steps in the modular reductions are reduced to L, where L serial steps, where w is a digit size in bits, and L is a number of digits of operands=[k/w].

Systems and methods for end-to-end encryption of a web browsing process are described herein. A web query is encrypted at a client using a homomorphic encryption scheme. The encrypted query is sent to a server where the encrypted query is evaluated over web content to generate an encrypted response without decrypting the encrypted query and without decrypting the response. The encrypted response is sent to the client where it is decrypted to obtain the results of the query without revealing the query or results to the owner of the web content, an observer, or an attacker.

Systems and methods for end-to-end encryption and dynamic resizing and encoding into grouped byte channels are described herein. A query is homomorphically encrypted at a client using dynamic channel techniques. The encrypted query is sent without a private key to a server for evaluation over target data to generate encrypted response without decrypting the encrypted query. The result elements of the encrypted response are grouped, co-located, and dynamically resized and encoded into grouped byte channels using the dynamic channel techniques, without decrypting the encrypted query or the encrypted response. The encrypted response is sent to the client where the client uses the private key and channel extraction techniques associated with the dynamic channel techniques to decrypt and perform channel extraction on the encrypted response to obtain the results of the query without revealing the query or results to a target data owner, an observer, or an attacker.