Various embodiments of the invention implement countermeasures designed to withstand attacks by potential intruders who seek partial or full retrieval of elliptic curve secrets by using Various embodiments of the invention implement countermeasures designed to withstand attacks by potential intruders who seek partial or full retrieval of elliptic curve secrets by using known methods that exploit system vulnerabilities, including elliptic operation differentiation, dummy operation detection, lattice attacks, and first real operation detection. Various embodiments of the invention provide resistance against side-channel attacks, such as simple power analysis, caused by the detectability of scalar values from information leaked during regular operation flow that would otherwise compromise system security. In certain embodiments, system immunity is maintained by performing elliptic scalar operations that use secret-independent operation flow in a secure Elliptic Curve Cryptosystem.

A single architected instruction to perform scalar multiplication for cryptographic operations is obtained. The instruction is executed, and the executing includes determining a scalar multiply function of a plurality of scalar multiply functions supported by the instruction to be performed. Input for the scalar multiply function is obtained, and the input includes at least one source component and a scalar value. The scalar multiply function is performed using the input to provide an output to be used in a cryptographic operation.

A computer processing system and method for reducing memory footprint that includes initiating, through at least one computer processor, a cryptography session utilizing an -degree isogeny arithmetic computation having chained computations therein. The cryptography session includes implementing a first iteration cycle, of a plurality of iteration cycles, and a implementing a remaining amount of the plurality of iteration cycles, each of the plurality iteration cycles computing isogenies using a compressed Z value to complete the -degree isogeny arithmetic computation. The first iteration cycle includes individually computing a plurality of sequentially occurring pivot points within the chained computations, implementing a Co-Z algorithm within the plurality of sequentially occurring pivot points to compute and store the compressed Z value on one of the plurality of temporary registers and computing a first isogeny of the -degree isogeny arithmetic computations using the compressed Z value.

A method of performing finite field addition and doubling operations in an elliptic curve cryptography (ECC) authentication scheme as a countermeasure to side-channel attack. The addition and doubling operations are executed using atomic patterns that involve the same sequence and number of operation types, so that the noise consumption and electromagnetic emanation profile of circuitry performing the operations is identical regardless of operation. A subtraction operation using such an atomic pattern is also disclosed.

Computer-implemented methods for locking a blockchain transaction based on undetermined data are described. The invention is implemented using a blockchain network. This may, for example, be the Bitcoin blockchain. A locking node may include a locking script in a blockchain transaction Node to lock a digital asset. The locking script includes a public key for a determined data source and instructions to cause a validating node executing the locking script to verify the source of data provided in an unlocking script by: a) generating a modified public key based on the public key for the determined data source and based on data defined in the unlocking script; and b) evaluating a cryptographic signature in the unlocking script based on the modified public key. The blockchain transaction containing the locking script is sent by the locking node to the blockchain network. The lock may be removed using a cryptographic signature generated from a private key modified based on the data.

A method for providing Cheon-resistance security for a static elliptic curve Diffie-Hellman cryptosystem (ECDH), the method including providing a system for message communication between a pair of correspondents, a message being exchanged in accordance with ECDH instructions executable on computer processors of the respective correspondents, the ECDH instructions using a curve selected from a plurality of curves, the selecting including choosing a range of curves; selecting, from the range of curves, curves matching a threshold efficiency; excluding, within the selected curves, curves which may include intentional vulnerabilities; and electing, from non-excluded selected curves, a curve with Cheon resistance, the electing comprising a curve from an additive group of order q, wherein q is prime, such that q1=cr and q+1=ds, where r and s are primes and c and d are integer Cheon cofactors of the group, such that cd48.

Various embodiments of the invention implement countermeasures designed to withstand attacks by potential intruders who seek partial or full retrieval of elliptic curve secrets by using known methods that exploit system vulnerabilities, including elliptic operation differentiation, dummy operation detection, lattice attacks, and first real operation detection. Various embodiments of the invention provide resistance against side-channel attacks, such as sample power analysis, caused by the detectability of scalar values from information leaked during regular operation flow that would otherwise compromise system security. In certain embodiments, system immunity is maintained by performing elliptic scalar operations that use secret-independent operation flow in a secure Elliptic Curve Cryptosystem.

Elliptic curve cryptographic schemes performed between a pair of cryptographic correspondent computing devices. In an aspect, a first entity generates a first basis point in a first selected basis being, either a first basis (A) or a second basis (B), and performs a first key generation in the selected basis. A second entity receives the public key and determines the product of a predetermined scalar in a second selected basis being either the first basis (A) or the second basis (B) and one of the first auxiliary points. If the product is an identity point, performs second key generation in the second selected basis, otherwise performing second key generation in either of the first basis (A) or the second basis (B). A common key is generated using the private keys and public keys. In another aspect, a scheme is performed symmetrically between two entities to generate a common key.

The invention relates to a cryptographic processing method comprising multiplication of a point P of an elliptic curve on a Galois field by a scalar k, the multiplication comprising steps of: storing, in a first register, a zero point of the Galois field, executing a loop comprising at least one iteration comprising steps of: selecting a window of w bits in the non-signed binary representation of the scalar k, w being a predetermined integer independent of the scalar k and strictly greater than 1, calculating multiple points of P being each associated with a bit of the window and of the form 2.sup.iP, adding or not in the first register of multiple points stored, depending of the value of the bit of the window with which the multiple points are associated, wherein the loop ends once each bit of the non-signed binary representation of the scalar k has been selected, returning a value stored in the first register. If all the bits of the window selected during an iteration of the loop are zero, the iteration comprises at least one dummy execution of the addition function, and/or if all the bits of the window during an iteration of the loop are non-zero, the multiple points to be added in the first register during the step are determined from a non-adjacent form associated with the window.

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.