In this application, example methods for performing quantum Montgomery arithmetic are disclosed. Additionally, circuit implementations are disclosed for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. This application also shows that elliptic curve discrete logarithms on an elliptic curve defined over an n-bit prime field can be computed on a quantum computer with at most 9n+2 log.sub.2(n)+10 qubits using a quantum circuit of at most 512n.sup.3 log.sub.2(n)+3572n.sup.3 Toffoli gates.

An apparatus and method for performing operation being secure against side channel attack are provided. The apparatus and method generate values equal to values obtained through an exponentiation operation or a scalar multiplication operation of a point using values extracted from previously generated parameter candidate value sets and an operation secure against side-channel attack, thereby improving security against side-channel attack without degrading performance.

An apparatus and method for performing operation being secure against side channel attack are provided. The apparatus and method generate values equal to values obtained through an exponentiation operation or a scalar multiplication operation of a point using values extracted from previously generated parameter candidate value sets and an operation secure against side-channel attack, thereby improving security against side-channel attack without degrading performance.

An apparatus and method for performing operation being secure against side channel attack are provided. The apparatus and method generate values equal to values obtained through an exponentiation operation or a scalar multiplication operation of a point using values extracted from previously generated parameter candidate value sets and an operation secure against side-channel attack, thereby improving security against side-channel attack without degrading performance.

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.

The present invention relates to a method for electronic signing of a document with a predetermined secret key (x), the method being characterized in that it comprises the implementation of steps of: (a) Drawing a pair formed by a first internal state (s.sub.1.sup.i) and a white-box implementation (WB.sub.i) of a modular arithmetic operation, from among a set of predetermined pairs ({(s.sub.1.sup.i,WB.sub.i)}.sub.i[0,n-1]) each for one nonce (k.sub.i), said first internal state (s.sub.1.sup.i) being a function of the nonce (k.sub.i) and said modular arithmetic operation being a function of the first internal state (s.sub.1.sup.i), of the nonce (k.sub.i) and of the secret key (x); (b) Determining a second internal state (s.sub.2.sup.i) by application of said drawn white-box implementation (WB.sub.i) to a condensate of the document obtained via a given hash function; (c) Generating an electronic signature of the document from the first internal state (s.sub.1.sup.i) of the drawn pair and from the second determined internal state (s.sub.2.sup.i), and deleting the drawn pair of said set of pairs ({(s.sub.1.sup.i,WB.sub.i)}.sub.i[0,n-1]).

Systems and methods for efficient fixed-base multi-precision exponentiation are disclosed herein. An example method includes applying a multi-precision exponentiation algorithm to a base number, the multi-precision exponentiation algorithm comprises a pre-generated lookup table used to perform calculations on the base number, the pre-generated lookup table comprising pre-calculated exponentiated values of the base number.

Methods and systems for masking certain cryptographic operations in a manner designed to defeat side-channel attacks are disclosed herein. Squaring operations can be masked to make squaring operations indistinguishable or less distinguishable from multiplication operations. In general, squaring operations are converted into multiplication operations by masking them asymmetrically. Additional methods and systems are disclosed for defeating DPA, cross-correlation, and high-order DPA attacks against modular exponentiation.

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 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.