Disclosed herein are a finite-field division operator, an elliptic curve cryptosystem having the finite-field division operator, and a method for operating the elliptic curve cryptosystem. The method for operating an elliptic curve cryptosystem may include, setting, by a key setting unit, a length of a key of a cryptographic algorithm, generating, by the key setting unit, first setup information that indicates a number of words corresponding to the key length, and generating, by the key setting unit, second setup information that indicates a number of repetitions of an operation by a finite-field division operator corresponding to the key length.

In one embodiment, it is proposed a method of cryptographic processing of data, the method being executed by an electronic device, and comprising obtaining at least two points belonging to a same elliptic curve defined on an algebraic structure being a finite ring, each point being represented by at least two coordinates. The method is remarkable in that it comprises: obtaining a parameterization of an isomorphism between said elliptic curve and another elliptic curve, said parameterization defining some configuration parameters, each configuration parameter having a range of possible values; determining in function of values of coordinates of said at least two points said configuration parameters, delivering determined configuration parameters; and obtaining coordinates of another point corresponding to an image of an addition of said at least two points through said isomorphism, said another point belonging to said another elliptic curve, and said obtaining being performed without an inversion operation in said algebraic structure, due to said determined configuration parameters.

Systems and methods described herein relate to techniques that allow for multiple parties to jointly generate or jointly agree upon the parameters for generation of a smart contract, such as a verification key. Execution of the smart contract may be performed by a third party, for example, a worker node on a blockchain network. Techniques described herein may be utilised as part of a protocol in which parties of a smart contract share powers of a secret in a manner that allows each party to determine an identical common reference string, agree on parameters for a smart contract, agree and/or make proportionate contributions the smart contract, and combinations thereof. The smart contract may be published to a blockchain network (e.g., Bitcoin Cash). The protocol may be a zero-knowledge protocol.

A cryptographic data processing method, implemented in an electronic device including a processor, the method including steps of providing a point of an elliptic curve in a Galois field, and a whole number, and of calculating a scalar product of the point by the number, the coordinates of the point and the number having a size greater than the size of words that may be processed directly by the processor, the scalar multiplication of the point by the number including steps of: storing scalar multiples of the point multiplied-by the number 2 raised to a power belonging to a series of whole numbers, setting a resulting point for each non-zero bit of the first number, adding the resulting point and one of the stored multiple points, and providing at the output of the processor the resulting point as result of the scalar product.

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 q−1=cr and q+1=ds, where r and s are primes and c and d are integer Cheon cofactors of the group, such that cd≦48.

A computer system module(s) substitutes a double scalar multiplication, used for signature verification in an encryption/decryption system, for two single scalar multiplications. The modules verify a group equation defined by [S]B=R+[k]A′ of the encryption/decryption system, where S is an integer characterized by the signature, K is an integer generated by a message being encrypted, B is a base point on the elliptic curve, R is a point on the elliptic curve and characterized by the signature, and A′ is a public key. The modules optionally rearrange the group equation to [S]B+[−k]A′=R, and convert it to [S]B+[n−k]A′=R, where n is the order of the base point. The modules determine a joint sparse form for the integers S and n−k and apply the Shamir's algorithm to the joint sparse form to verify the group equation.

Embodiments are directed to countermeasures for side-channel attacks on protected sign and key exchange operations. An embodiment of storage mediums includes instructions for commencing a process including an elliptic curve scalar multiplication (ESM) operation including application of a secret scalar value; splitting the secret scalar value into two random scalar values; counting a number of leading ‘0’ bits in the scalar value and skipping the number of leading ‘0’ bits in processing; performing an ESM iteration for each bit of the secret scalar value beginning with a most significant ‘1’ bit of the scalar value including a Point Addition operation and a Point Double operation for each bit on randomized points; performing ESM operation dummy iterations equal to the number of leading ‘0’ bits; and returning an output result for the ESM operation.

Disclosed are devices, systems and methods for performing secure transactions in an aircraft are disclosed. Embodiments of the disclosed technology enable low cost carriers to provide payment verification for on-board purchases via the in-flight entertainment system. An exemplary method for performing secure transactions in an aircraft includes transmitting, by a user device in the aircraft using a wireless protocol, a first authentication factor and a request for one or more on-board services; receiving, from an on-board transceiver using the wireless protocol, an authentication token (a) comprising a one-time code and (b) encrypted using an asymmetric cryptographic algorithm; transmitting, using the asymmetric cryptographic algorithm, a second authentication factor comprising (a) the authentication token and (b) a text message transmitted from the user device; and receiving a confirmation of a delivery of the one or more on-board services.

Disclosed are a system and method for calculating elliptic curve cryptography scalar multiplication using an FPGA (Field Programmable Gate Array), the system and method scheduling calculation, which is used in a Montgomery ladder Algorithm, and enabling efficient calculation through an improved modular arithmetic calculation method. The system for calculating elliptic curve cryptography (ECC) scalar multiplication using an FPGA includes: a scheduler implementing Montgomery ladder step calculation in a pipeline structure; a pipeline modular adder/subtractor implementing n-bit modular addition in a d-stage pipeline structure; and a modular multiplier implementing n-bit modular multiplication in a 10-stage pipeline structure up to maximum 256 bits.

A calculation is performed on a first number and a second number. For each bit of the second number a first function is performed. The first function inputs include contents of a first register, contents of a second register and the first number. A result of the first function is placed in a third register. For each bit of the second number, a second function is performed which has as inputs contents of the third register and the contents of a selected one of the first and the second register according to a state of a current bit of the second number. A result of the second function is stored in the selected one of the first and second register.