198 results about "Scalar multiplication" patented technology

A method and an apparatus capable of realizing at a high speed an elliptic curve cryptography in a finite field of characteristic 2, in which the elliptic curve is given by y²+xy=x³+ax²+b (b≠0) and an elliptic curve cryptography method which can protect private key information against leaking from deviation information of processing time to thereby defend a cipher text against a timing attack and a differential power analysis attack are provided. To this end, an arithmetic process for executing scalar multiplication arithmetic d(x, y) a constant number of times per bit of the private key d is adopted. Further, for the scalar multiplication d(x, y), a random number k is generated upon transformation of the affine coordinates (x, y) to the projective coordinates for thereby effectuating the transformation (x, y)→[kx, ky, k] or alternatively (x, y)→[k²x, k³y, k]. Thus, object for the arithmetic is varied by the random number (k).

In scalar multiplication method in which a point on an elliptic curve is randomized, but yet scalar multiplication can be calculated by the computational cost as much as that without randomization, an operation is carried out upon a point randomized and a point not randomized in a scalar multiplication method to calculate a scalar-multiplied point from a scalar value and a point on an elliptic curve. The result of the operation is randomized while the computational cost becomes as much as that without randomization.

The method for elliptic curve scalar multiplication may provide several countermeasures to protect scalar multiplication of a private key k by a point P to produce the product kP from power analysis attacks. First, the private key, k, is partitioned into a plurality of key partitions, which are processed in a random order, the resulting points being accumulated to produce the scalar product kP. Second, in each partition, the encoding is randomly selected to occur in binary form or in Non-Adjacent Form (NAF), with the direction of bit inspection being randomly assigned between most-to-least and least-to-most. Third, in each partition, each zero in the key may randomly perform a dummy point addition operation in addition to the doubling operation. The method may be implemented in software, smart cards, circuits, processors, or application specific integrated circuits (ASICs) designed to carry out the method.

An encryption device (FIG. 15) performs elliptic curve encryption using a secret key. The encryption device includes: operation means (ECDBL, ECADD) for performing scalar multiplication of a point on an elliptic curve; storage (T[0]-T[2]) having a plurality of data storing areas; and means (SEL) for determining, in accordance with a bit sequence of a given value (d) and with a random value (RNG), an address of one of the plurality of data storage areas that is to be coupled to the operation means for each scalar multiplication.

A randomly selected point on an elliptic curve is set as the initial value of a variable and calculation including a random point value is performed in an algorithm for calculating arbitrary scalar multiple operation on an elliptic curve when scalar multiplication and addition on an elliptic curve are defined, then a calculation value obtained as a result of including a random point is subtracted from the calculation result, whereby an intended scalar multiple operation value on an elliptic curve is determined.

A cryptographic system and method for encrypting input data, in which an example system includes a table calculator configured to calculate table values composed of one of scalar multiplication values by Elliptic Curve (EC) operation, or exponentiation values by modular exponentiation operation, based on input data and the number of a portion of bits of each of secret keys. The table calculator may output one of scalar multiplication values or exponentiation values corresponding to a window that includes given bits of each of the secret keys from among the calculated table values. A logic circuit may be configured to output encrypted data by accumulating the output scalar multiplication values or by performing involution on the output exponentiation values.

The method for elliptic curve scalar multiplication may provide several countermeasures to protect scalar multiplication of a private key k by a point P to produce the product kP from power analysis attacks. First, the private key, k, is partitioned into a plurality of key partitions, which are processed in a random order, the resulting points being accumulated to produce the scalar product kP. Second, in each partition, the encoding is randomly selected to occur in binary form or in Non-Adjacent Form (NAF), with the direction of bit inspection being randomly assigned between most-to-least and least-to-most. Third, in each partition, each zero in the key may randomly perform a dummy point addition operation in addition to the doubling operation. The method may be implemented in software, smart cards, circuits, processors, or application specific integrated circuits (ASICs) designed to carry out the method.

In scalar multiplication method using a Montgomery-type elliptic curve, a high-speed elliptic curve calculation device that can use effectively a table that stores coordinates of a certain scalar multiple points like points multiplied by exponentiation of two to a certain point G and so forth. An elliptic curve calculation device 200 that receives an arbitrary integer k of n bits and outputs scalar-multiplied points against a point G on a Montgomery-type elliptic curve E on an infinite field F that is given in advance comprises: a calculation procedure generation unit 210 that generates a calculation procedure that addition on the elliptic curve E with either of G, 2*G, 22*G, . . . , 2n-1*G as the first addition element is repeated and a scalar multiplication unit 220 that calculates the scalar-multiplied points k*G by repeating addition on the elliptic curve E, referring to a table memorizing unit 220b that stores values (coordinates) of exponentiation of two against the point G and complying with the generated calculation procedure.

A method for implementing an elliptic curve or discrete logarithm cryptosystem on inexpensive microprocessors is disclosed which provides for advantageous finite field computational performance on microprocessors having limited computational capabilities. The method can be employed with a variety of commercial and industrial imbedded microprocessor applications such as consumer smart cards, smart cards, wireless devices, personal digital assistants, and microprocessor controlled equipment. In one embodiment, a Galois Field (GF) implementation based on the finite field GF((2⁸−17)¹⁷) is disclosed for an Intel 8051 microcontroller, a popular commercial smart card microprocessor. The method is particularly suited for low end 8-bit and 16-bit processors either with or without a coprocessor. The method provides for fast and efficient finite field multiplication on any microprocessor or coprocessor device having intrinsic computational characteristics such that a modular reduction has a greater computational cost than double precision, long number additions or accumulations. The disclosed method offers unique computational efficiencies in requiring only infrequent subfield modular reduction and in employing an adaptation of Itoh and Tsujii's inversion algorithm for the group operation. In one embodiment, a core operation for a signature generation, an elliptic curve scalar multiplication with a fixed point, is performed in a group of order approximately 2¹³⁴ in less than 2 seconds. In contrast to conventional methods, the method does not utilize or require curves defined over a subfield such as Koblitz curves.

A system and method for achieving secure and fast computation in hyperelliptic cryptography is realized. Fast scalar multiplication is achieve by executing computing operations including halving as computing processing in scalar multiplication with respect to a divisor D in hyperelliptic curve cryptography. For example, computing operations including halving are executed in scalar multiplication with respect to a divisor D on a hyperelliptic curve of genus 2 in characteristic 2 having h(x)=x²+x+h₀, f₄=0 as parameters, a hyperelliptic curve of genus 2 in characteristic 2 having h(x)=x²+h₁x+h₀, f₄=0 as parameters, or a hyperelliptic curve of genus 2 in characteristic 2 having h(x)=x as a parameter. Further, reduced complexity and faster computation are realized through the application of a table that records which of k₁, k₁′, (k₀, k₀′) is correct on the basis of a computed value of [½ⁱD] with respect to a fixed divisor D, and through a reduction in the number of inversion operations.

A method and system are provided for atomicity for elliptic curve cryptosystems (ECC-systems). The method includes a side channel atomic scalar multiplication algorithm using mixed coordinates. The algorithm including repeating a sequence of field operations for each elliptic curve addition or doubling operation to provide an atomic block, wherein an atomic block appears equivalent by side-channel analysis. The mixed coordinates are chosen based on a ratio of I/M where I and M are the time required to execute an inversion and a multiplication in the ground field respectively. If the I/M ratio is less than 60, a mixture of affine and Jacobian coordinates are used during scalar multiplication. If the I/M ratio is 60 or more, a mixture of Chudnovsky-Jacobian and Jacobian coordinates are used during scalar multiplication. The method is optimized for elliptic curves over F_p defined by an equation of the form y²=x³+ax+b, where a, b ε F_p, having a=−3.

The invention provides a quasi-cyclic LDPC serial encoder based on ring shift left, which includes c generator polynomial lookup tables of all cyclic matrix generator polynomials in a prestored generated matrix, c b-bit binary multipliers for performing scalar multiplication to information bit and generator polynomials, c b-bit binary adders for performing modulo-2 adding to product and content of shifting registers, and c b-bit shifting registers for storing the sum of ring shift left for 1 bit, and finally, the checking data is stored in the c shifting registers. The serial encoder provided by the invention has the advantages of few registers, simple structure, low power consumption, low cost and the like.

An apparatus and a method for performing a hyperelliptic curve cryptography process at a high speed in a highly secure manner are provided. A base point D is produced such that the base point D and one or more of precalculated data in addition to the base point used in a scalar multiplication operation based on a window algorithm are degenerate divisors with a weight smaller than genus g of a hyperelliptic curve. An addition operation included in the scalar multiplication operation based on the window algorithm is accomplished by performing an addition operation of adding a degenerate divisor and a non-degenerate divisor, whereby a high-speed operation is achieved without causing degradation in security against key analysis attacks such as SPA.

The invention provides a quasi-cyclic matrix serial multiplier based on rotate left. The quasi-cyclic matrix serial multiplier is used for implementing multiplication of a vector m and a quasi-cyclic matrix F in QC-LDPC (quasi-cyclic low-density parity-check) approximate lower triangular encodings. The multiplier comprises u generated polynomial lookup tables, u b-bit binary multipliers, u b-bit binary summators and u b-bit shift registers. The generated polynomial lookup tables are used for pre-storing quasi-cyclic matrix generated polynomial in the matrix F. The b-bit binary multipliers are used for performing scalar multiplication of the vector m data bit and the generated polynomial. The b-bit binary summators are used for performing modulo-2 adding of products and shift register contents. The b-bit shift registers are used for storing sums of 1-bit movements subjected to rotate left. The quasi-cyclic matrix serial multiplier has the advantages of small number of registers, simple structure, low power consumption and cost and the like.

A method for multiplying, at an execution unit of a processor, two complex numbers in which all four scalar multiplications, concomitant to multiplying two complex numbers, can be performed in parallel. A real part of a first complex number is multiplied at the execution unit by a real part of a second complex number to produce a first part of a real part of a third complex number. An imaginary part of the first complex number is multiplied at the execution unit by an imaginary part of the second complex number to produce a second part of the real part of the third complex number. A first arithmetic function is performed at the execution unit between the first part of the real part of the third complex number and the second part of the real part of the third complex number. The imaginary part of the first complex number is multiplied at the execution unit by the real part of the second complex number to produce a first part of an imaginary part of the third complex number. The real part of the first complex number is multiplied at the execution unit by the imaginary part of the second complex number to produce a second part of the imaginary part of the third complex number. A second arithmetic function is performed at the execution unit between the first part of the imaginary part of the third complex number and the second part of the imaginary part of the third complex number.

The invention discloses a random point generation method suitable for elliptic curve cryptography (ECC) safety protection. The method comprises the following steps: (1) finding a fixed point P on a certain elliptic curve except an infinite point; (2) selecting a random number r; (3) calculating scalar multiplication of r and P to obtain a calculation result Q; (4) determining whether Q is the infinite point, if yes, returning to the step (2), and otherwise, executing the step (5); and (5) obtaining the random point Q on the certain elliptic curve. The method can obviate the large amount of modular multiplication by obviating the evolution of large numbers or the operation for solving a quadratic equation with one unknown. Therefore, the method provided by the invention can greatly improve the operation speed and reduce the time for generating the random point on the elliptic curve.

The invention relates to an acentric digital currency transaction method based on public and private key pair derivation; a receiver can scan on a public account book to calculate collection public and private key pairs according to the transfer time, thus reducing data storage; each transaction forms a new collection address, thus enhancing the digital currency anonymous property; only one time elliptical curve scalar multiplication calculation is added on the computational complexity in an implicit certificate generation process, thus easily reducing stronger anonymous property; a transaction initiator actively forms a new collection address in the process instead of using the new collection address issued by the receiver; each person only processes a long term public key, and new publicand private key pairs can be calculated according to the public messages on the public account book and the long term public and private key pairs, thus reducing the public and private key pair storage space; transaction graph analysis is hard to receive.

A method performs scalar multiplication of points on an elliptic curve by a finite expandable field K of a first field F_p of a p>3 characteristic, wherein said characteristic p has low Hamming weight and the expandable field has a polynomF(X)+X^d−2 of order d in the polynomial representation thereof.

An elliptic curve scalar multiplication apparatus stores a prime number p and information of a first point, the prime number p defining a field of definition F_p, which defines a first curve, which is a Weierstrass form elliptic curve, and expressed as p=p₀+p₁c+ . . . +p₁cⁿ⁻¹, (where c equals 2^f and f is an integer equal to or larger than 1 that is units of breaking data into pieces in multiple-precision integer arithmetic executed by the elliptic curve scalar multiplication apparatus), calculates a Montgomery constant k₀, work, and h₁, executes doubling of a second point, which is calculated from the first point, by Montgomery multiplication that uses k₀, work, and h₁, adds a third point and fourth point, which are calculated from the first point, by Montgomery multiplication that uses k₀, work, and h₁; and calculates a scalar multiple of the first point, based on a result of the doubling and the addition.