KR20210026410A - Computing Apparatus and Method for Hardware Implementation of Public-Key Cryptosystem Supporting Elliptic Curves over Prime Field and Binary Field - Google Patents

Abstract

The present invention relates to an operation device and an operation method for hardware implementation of a public-key encryption system supporting an elliptic curve on a prime field and a binary field. More specifically, the present invention relates to an operation device and an operation method capable of implementing hardware of a public key encryption system based on an elliptic curve such as EC-DSA, EC-ElGamal, and EC-DH while supporting elliptic curves of various sizes on a prime field and a binary field. To this end, the present invention comprises: a word-based finite field operator including a modular adder/subtractor having a size of ω-bit and a word-based Montgomery modular multiplier having a size of ω-bit; a memory for storing an intermediate result value of an elliptic curve domain parameter and a word-based finite field operation; a register block for storing a value for setting a motion mode and for storing an intermediate result value of the word-based finite field operation; and a motion mode control unit for controlling the word-based finite field operator, memory and register block. According to the present invention, the present invention can be applied in implementation of various public key encryption protocols, thereby exhibiting usability and cost-reducing effects.

Description

Computing Apparatus and Method for Hardware Implementation of Public-Key Cryptosystem Supporting Elliptic Curves over Prime Field and Binary Field}

The present invention relates to a computing device and method for hardware implementation of a public key cryptographic system that supports elliptic curves on prime numbers and binary bodies, and more specifically, to support elliptic curves of various sizes on prime numbers and binary bodies, The present invention relates to a computing device and method capable of hardware implementation of an elliptic curve-based public key cryptosystem such as EC-DSA, EC-ElGamal, and EC-DH.

With the commercialization of 5G mobile communication technology, the amount of information transmitted and devices connected through wireless networks is increasing exponentially.In particular, the Internet of Things, self-driving cars, unmanned aerial vehicles (drones), smart homes, smart The importance of information security technology to protect information and devices from cyber attacks as devices and systems that are connected to wired and wireless networks such as health devices, personal information terminals, electronic wallets, and virtual currency that transmit, store, and process information are increasing. This is getting bigger.

For information security functions such as confidentiality, authentication, and integrity verification, a secret key (symmetric key) encryption method, a public key encryption method, and a cryptographic hash function are used. RSA (Rivest, Shamir, Adleman) encryption and Elliptic Curve Cryptography are widely used as public key encryption methods for digital signatures, key exchange, and authentication. The RSA encryption method has a disadvantage in that it requires a very large key length of 1024-bit or 2048-bit, and thus requires large memory and hardware requirements. On the other hand, elliptic curve encryption has the advantage of being advantageous in hardware and software implementation by having a security strength similar to that of RSA with a key length of several hundred bits, and recently, security for autonomous vehicles, drones, electronic wallets, virtual currency, etc. For this, the use of the elliptic curve public key encryption method is rapidly expanding.

Elliptic Curves Elliptic curves of various sizes (key lengths) are defined in the standard documents of the National Institute of Standards and Technology (NIST) and SEC-2, and based on this, the elliptic curve-based electronic signature Various public key cryptographic protocols are being implemented, such as EC-DSA, a protocol, Diffie-Hellman (EC-DH), an elliptic curve-based key exchange protocol, and EC-ElGamal, a message encryption and decryption protocol based on an elliptic curve.

Elliptic curve ciphers are classified into an elliptic curve cipher on a minority body and an elliptic curve cipher on a binary body according to the applied finite field, and various sizes of sieves are used. In FIPS 186-4, NIST's digital signature standard, there are five (192, 224, 256, 384, 521 bits) sized ellipses on a minor body and five (163, 233, 283, 409, 571 bits) sizes on a binary body. The curve is defined, and in SEC-2, there are 8 types of hydrophobic bodies (secp192k1, secp192r1, secp224k1, secp224r1, secp256k1, secp256r1, secp384r1, secp521r1) and 12 types of binary bodies (sect163k1, sect163r1, sect163r2, sect2331, sect163r2, sect2331, sect233k1 , sect283k1, sect283r1, sect409k1, sect409r1, sect571k1, sect571r1) Elliptic curves are defined.

The elliptic curve cipher has different sizes of finite fields and elliptic curves according to application fields and systems, and domain parameters such as elliptic curve equation coefficients, generation point values, and modular values vary accordingly. In order to implement the dedicated hardware for elliptic curve encryption, flexibility of the operation device and operation method is required to support various elliptic curves specified in domestic and foreign standards.

In addition, point scalar multiplication (PSM), point addition (PA), point subtraction (PS) for implementation of elliptic curve encryption application protocols such as EC-DSA, EC-ElGamal, and EC-DH ), modular addition (MA), modular subtraction (MS), modular multiplication (MM), modular division, as well as elliptic point operation of point doubling (PD). MD), modular inversion (MI) finite field modular arithmetic operation is required. In addition, for encryption of EC-ElGamal, a process of mapping message data to a point on an elliptic curve is required, and for this, a modular square root (MSQRT) calculation is required.

Accordingly, an efficient operation device and operation that supports decimal and binary elliptic curves of various sizes using one hardware device, and provides elliptic curve point operation, finite field modular arithmetic operation, and elliptic curve point mapping function of messages. There is a need to develop a method.

The present invention was conceived to solve the above problems, and through a word-based finite field computing device and memory, and an efficient control method for them, ellipses of various key lengths defined in the standard document of the US National Institute of Standards and Technology and Its purpose is to provide a computing device and method capable of hardware implementation of an elliptic curve public key cryptosystem such as an elliptic curve-based electronic signature (EC-DSA), EC-ElGamal, and EC-DH (Diffie-Hellman) by supporting curves. .

The present invention has the following features to achieve the above object.

a word-based finite field operator including a ω-bit sized modular add/subtracter and an ω-bit sized word-based Montgomery modular multiplier; A memory for storing an elliptic curve domain parameter and an intermediate result value of a word-based finite field operation; A register block that stores a value for setting an operation mode and stores an intermediate result value of a word-based finite field operation; And an operation mode control unit for controlling the word-based finite field operator, memory, and register block.

Here, the operation modes stored in the register block and controlled by the operation mode controller include an elliptic curve point operation mode and a finite field modular operation mode.

계산모드와, 몽고메리 도메인의 연산결과를 정수 도메인으로 변환하는 역매핑모드와, 좌표계 변환을 위한 연산과, 몽고메리 모듈러 곱셈 과정에서 합동(congruence)을 이용한 축약에 사용되기 위한

값의 계산모드와, 임의의 메시지 값을 타원곡선 상의 점으로 매핑하는 과정에서 필요한 모듈러 제곱근 계산모드를 포함한다. In addition, the operation mode is for mapping data to the Montgomery domain.

A calculation mode, an inverse mapping mode that converts the operation result of the Montgomery domain into an integer domain, an operation for transforming the coordinate system, and a reduction using congruence in the Montgomery modular multiplication process.

It includes a value calculation mode and a modular square root calculation mode required in the process of mapping an arbitrary message value to a point on an elliptic curve.

In addition, the word-based Montgomery modular multiplier includes an ω-bit sized dual field multiplier that selectively calculates multiplication on a prime number and a multiplication on binary, a dual field adder of ω-bit size, and a register of ω-bit size, It includes a 1-bit carry storage register, a 2-bit carry storage register, a 1-bit adder, and a plurality of selectors.

가 계산되고,

값과 데이터를 몽고메리 모듈러 곱셈하여 몽고메리 도메인으로 매핑하는 제2단계와; 동작모드에 따른 동작모드 제어부의 제어에 따라 타원곡선 점 연산(PSM, PA, PS, PD)과 유한체 모듈러 연산(MM, MD, MI, MA, MS)이 수행되는 제3단계와; 좌표계 변환 단계로, 소수체의 경우에는 자코비안(Jacobian) 좌표계에서 2차원 아핀(affine) 좌표계로, 이진체의 경우에는 로페즈-다합(Lopez-Dahab) 좌표계에서 2차원 아핀 좌표계로 변환이 수행되는 제4단계와; 몽고메리 도메인에서 계산된 연산결과에 포함되어 있는

을 제거하기 위해 리매핑되는 제5단계와; 상기 제5단계의 결과데이터가 출력되는 제6단계;를 포함한다. In addition, a first step of inputting an elliptic curve domain parameter and data to a computing device through the computing device in a calculation method according to an embodiment of the present invention; As an initial parameter value according to an elliptic curve under the control of the operation mode controller, it is necessary to map data to the Montgomery domain for word-based Montgomery modular multiplication.

Is calculated,

A second step of performing a Montgomery modular multiplication of values and data to map them to a Montgomery domain; A third step of performing elliptic curve point calculations (PSM, PA, PS, PD) and finite field modular calculations (MM, MD, MI, MA, MS) according to the control of the operation mode controller according to the operation mode; In the case of a coordinate system conversion step, in the case of a minor body, the conversion is performed from a Jacobian coordinate system to a two-dimensional affine coordinate system, and in the case of a binary body, a two-dimensional affine coordinate system is performed. A fourth step; Included in the calculation result calculated in the Montgomery domain

A fifth step of being remapped to remove And a sixth step of outputting the result data of the fifth step.

값과, 몽고메리 곱셈 과정에서 합동(congruence)을 이용한 축약 연산에 사용되는

를 계산하도록 구성된다. Here, the computing device supports elliptic curves having multiple shapes through repetitive calculations according to the bit size of the finite field calculator, and after receiving a modular value p that varies according to the elliptic curve from the outside, Montgomery's modular multiplication and finite field Required for calculation

Value and the congruence used in the Montgomery multiplication process.

Is configured to calculate.

In addition, the computing device is configured to calculate a modular square root operation required in the process of mapping message data to points on the elliptic curve for EC-ElGamal encryption based on an elliptic curve.

According to the present invention, a finite field operator, a memory, and a method of processing various sizes of elliptic curve ciphers on a decimal field and a binary body with a single hardware through their operation control, and a modular arithmetic operation and a message without an additional computing device. By providing an elliptic curve point mapping method of, hardware complexity is reduced, and since it can be applied to the implementation of various public key cryptographic protocols based on elliptic curve encryption, it is effective in reducing utility and cost.

계산 과정의 슈도코드를 나타내는 도면이다.
도 5는 본 발명의 일실시예에 따라

계산 과정의 슈도코드를 나타내는 도면이다.
도 6은 본 발명의 다른 실시예에 따른 메시지 데이터 X를 타원곡선 상의 점 (X,Y)로 매핑하는 방법의 슈도코드의 나타내는 도면이다. FIG. 1 is a block diagram schematically showing an operation device for hardware implementation of a public key cryptographic system supporting elliptic curves on a prime number and a binary body according to an embodiment of the present invention.
2 is a block diagram showing the internal configuration of the word-based Montgomery modular multiplier of FIG. 1 according to an embodiment of the present invention.
3 is a diagram illustrating an operation pseudocode of a word-based Montgomery multiplier that supports decimal and binary characters according to an embodiment of the present invention.
4 is according to an embodiment of the present invention

It is a diagram showing the pseudocode of the calculation process.
5 is according to an embodiment of the present invention

It is a diagram showing the pseudocode of the calculation process.
6 is a diagram illustrating a pseudocode of a method of mapping message data X to points (X,Y) on an elliptic curve according to another embodiment of the present invention.

In order to explain the present invention and the operational advantages of the present invention and the object achieved by the implementation of the present invention, the following describes a preferred embodiment of the present invention and looks at with reference thereto.

First, terms used in the present application are only used to describe specific embodiments, and are not intended to limit the present invention, and expressions in the singular may include a plurality of expressions unless clearly differently in context. In addition, in the present application, terms such as "comprise" or "have" are intended to designate the existence of features, numbers, steps, actions, components, parts, or a combination thereof described in the specification, but one or more other It is to be understood that the presence or addition of features, numbers, steps, actions, components, parts, or combinations thereof, does not preclude the possibility of preliminary exclusion.

In describing the present invention, when it is determined that a detailed description of a related known configuration or function may obscure the subject matter of the present invention, a detailed description thereof will be omitted.

FIG. 1 is a block diagram schematically showing a computing device for hardware implementation of a public key cryptographic system supporting elliptic curves on a prime number and a binary body according to an embodiment of the present invention, and FIG. 2 is an embodiment of the present invention. 1 is a block diagram showing the internal configuration of the word-based Montgomery modular multiplier according to FIG.

계산 과정의 슈도코드를 나타내는 도면이며, 도 5는 본 발명의 일실시예에 따라

계산 과정의 슈도코드를 나타내는 도면이며, 도 6은 본 발명의 일실시예에 따른 메시지 데이터 X를 타원곡선 상의 점 (X,Y)로 매핑하는 방법의 슈도코드의 나타내는 도면이다. In addition, FIG. 3 is a diagram showing an operation pseudocode of a word-based Montgomery multiplier supporting decimal and binary characters according to an embodiment of the present invention, and FIG. 4 is a diagram illustrating an operation pseudocode according to an embodiment of the present invention.

It is a diagram showing a pseudocode of a calculation process, and FIG. 5 is a diagram showing a pseudocode of a calculation process.

Fig. 6 is a diagram showing a pseudocode of a calculation process, and FIG. 6 is a diagram showing a pseudocode of a method of mapping message data X to points (X,Y) on an elliptic curve according to an embodiment of the present invention.

Referring to the drawings, the computing device 100 for hardware implementation according to an embodiment of the present invention includes a modular adder/subtracter 11 having an ω-bit size and a word-based Montgomery modular multiplier 12 having an ω-bit size. A word-based finite field operator 10 that includes, a memory 20 that stores an elliptic curve domain parameter and an intermediate result value of a word-based finite field operation, and a word-based finite field operation that stores a value for setting an operation mode. A register block 30 storing the intermediate result value of the word-based finite field operator 10, the memory 20, and an operation mode control unit 40 that controls the register block 30.

Here, the word-based finite field operator 10 is composed of a ω-bit-sized modular adder/subtracter 11 and a ω-bit-sized word-based Montgomery modular multiplier 12, where ω is a register through machine language instructions or operations. It is the size of a word, which means a unit of data that can be replaced with, and is a natural number, preferably 16-bit, 32-bit, 48-bit, or 64-bit, and can be appropriately selected and implemented as needed. have.

((여기서 R=2^N이고, N은 타원곡선의 크기를 나타내는 비트 수, p는 소수체의 경우 모듈러 값, 이진체의 경우 기약다항식을 나타냄)과 워드 기반 유한체 연산의 중간 결과값을 저장한다.The memory 20 stores an elliptic curve coefficient value, a key value, a modular value (a decimal p in the case of a decimal number, a short polynomial in the case of a binary body), and the value of the generation point of the elliptic curve, and also Montgomery Modular Internally calculated to map the data to the Montgomery domain for multiplication.

(Where R = 2 ^N , N is the number of bits representing the size of the elliptic curve, p represents a modular value for a prime number, and a short polynomial for a binary body) and the intermediate result of word-based finite field operation is stored. do.

(여기서,

는 모듈러 값 p의 최하위 워드이고, ω는 워드의 크기)값을 저장하여 몽고메리 모듈러 곱셈 과정에서 합동을 이용한 축약에 사용되도록 한다. The register block 30 stores a value for setting the operation mode, and also internally calculated

(here,

Is the least significant word of the modular value p, and ω is the size of the word) so that it is used for contraction using congruence in the Montgomery modular multiplication process.

In addition, the key value stored in the memory 20 is read and stored in word units to be used for point operation and finite field modular operation, and intermediate result values of word-based finite field operation are also stored.

The operation mode controller 40 controls the operation of the word-based finite field operator 10, the memory 20, and the register block 30 according to the value of the operation mode setting register of the register block 30.

계산과 그 역매핑, 좌표계 변환을 위한 연산과, 몽고메리 모듈러 곱셈 과정에서 합동을 이용한 축약에 사용되기 위한

값의 계산과, 임의의 메시지 값을 타원곡선 상의 점으로 매핑하는 과정에서 필요한 모듈러 제곱근 계산 등을 포함한다.The operation mode according to the finite field operation device and control method according to an embodiment of the present invention includes elliptic curve point operation (PSM, PA, PS, PD) and finite field modular operation (MM, MD, MI, MA, MS). And also for mapping data to the Montgomery domain.

For use in calculations and inverse mapping, for coordinate system transformation, and for reduction using congruence in the Montgomery modular multiplication process.

It includes the calculation of the value and the calculation of the modular square root required in the process of mapping an arbitrary message value to a point on an elliptic curve.

An example of the setting value of the operation mode setting register is shown in [Table 1] below.

Various public key cryptographic protocols such as EC-DSA, an elliptic curve-based electronic signature, EC-DH (Diffie-Hellman), an elliptic curve-based key exchange protocol, and EC-ElGamal, an elliptic curve-based message encryption and decryption protocol. Implementation becomes possible.

Here, the elliptic curve point operation mode can implement point scalar multiplication (PSM), point addition (PA), point doubling (PD), and point subtraction (PS) operations on the elliptic curve used in the public key cryptographic protocol. , Selectively calculate the elliptic curve point operation on the decimal body and the elliptic curve point operation on the binary body.

In addition, the finite field modular operation mode can implement modular addition (MA), modular subtraction (MS), modular multiplication (MM), modular division (MD), and modular inverse (MI) operations used in public key cryptographic protocols.

At this time, the modular multiplication is performed by the word-based Montgomery modular multiplier 12 of the word-based finite field operator 10, and this word-based Montgomery modular multiplier 12 selectively calculates the multiplication on the prime number and the multiplication on the binary body. ω-bit sized dual field multiplier 12a, ω-bit sized dual field adder 12b, ω-bit sized register 12c, 1-bit carry storage register 12d, and 2- It comprises a bit carry storage register 12e, a 1-bit adder 12f, and a plurality of selectors 12g.

A process of performing an operation through the word-based Montgomery modular multiplier 12 is as shown in FIG. 3.

Looking at this, first, in Steps 1 to 4, before word-based Montgomery multiplication is performed, the finite field multiplication result S, the carry values C1 _-1 and C2 _-1 by the multiplication, and the carry values cy1 and cy2 by the addition are In step 1, in which all of the initial values are set to 0, the initial value is set at the start of the multiplication operation, and in steps 3 and 4, the initial value is set each time the i-loop is started.

In addition, in steps 7 and 18, the multiplication of the word a _i of the multiplicand and the word b _j of the multiplier a _i × b _j is calculated. Among the multiplication result values, the upper word C1 is used as a carry input in the (j+1)-loop operation, and the lower word P1 is used as a partial product of the j-loop.

In addition, in Steps 8 and 19, the operation on the prime number is the partial product P1 calculated in the j-loop of Step 8, the carry value C1 _j-1 of the multiplication calculated in the (j-1)-loop, and the carry value by the addition operation. Calculate the sum H of the partial products and the carry output cy1 by adding the sum _{s j} of the partial products calculated in cy1, (i-1)-loop.

In the operation on the binary body, the sum of partial products H is calculated by performing an XOR operation of _{P1, C1 j-1} and s _{j through step 19.}

와 부분 곱의 합 H를 곱하여 s_-1=0인 모듈러 합동 값 S를 만들기 위한 데이터 q_i 값을 계산한다. 계산된 q_i 값은 소수체 상의 연산에서는 단계 12 및 단계 13, 이진체 상의 연산에서는 단계 23 및 단계 24를 통해서 모듈러 합동 값 s_j-1이 계산된다.In addition, when step 9 to step 11 and step 20 to step 22 are the least significant word j = 0, data for modular congruence

By multiplying by the sum of the partial products H and s _{-1 = 0, the data q i} value is calculated to produce a modular congruent value S. The calculated q _{i value is a modular joint value s j-1} is calculated through steps 12 and 13 in an operation on a decimal body and steps 23 and 24 in an operation on a binary body.

Steps 12 and 13, and 23 and 24 generate a modular joint value S of _{s −1 = 0 by addition operation.} In the case of j=0, which is the least significant word, _{the process of s -1} = 0 in the operation on the decimal body is the same as (1) to (3) below, and in the operation on the binary body, the addition operation is processed as XOR.

(1) Step 10:

(2) Step 12:

(3) Step 13:

Meanwhile, in steps 15 and 26, the most significant word s _{m -1} of the result of the multiplication operation is calculated. s _m-1 has a size that is increased by 1-bit than that of a word. In an operation on a decimal body, a carry occurs due to addition, so the most significant bit of _{s m-1 s msb} can have a value of 0 or 1, and in an operation on a binary body, since carry by addition does not occur, s _msb = 0 to be.

In step 29, when the iteration of the i-loop is completed and the partial product addition is completed, the multiplication result S on the prime number and the modular value p are compared, and if S>p, the subtraction operation of S-p is performed. The modular reduction operation of the final multiplication result value is calculated by the modular adder/subtracter 11.

계산모드에 따른 수행과정을 설명하도록 한다. Meanwhile, to map the data to the Montgomery domain

The execution process according to the calculation mode will be described.

When calculating the data A, B in the modular multiplication algorithm the Montgomery multiplications for the modular value p, and a MM (A, B, p) = A × B × R -1 mod p, R -1 = 2 in the multiplication result ^{- Output is multiplied by N} (here, N represents the number of bits of the modular value p).

The ^{reason that the multiplication result is obtained in the form of multiplying R -1} is that the least significant word of the partial product addition result is made 0 using the congruence property, and the reduction operation is processed by removing it. Therefore, for modular multiplication by the Montgomery algorithm, as shown in [Table 2] below, the multiplication result by multiplying the operands A and B of the multiplication by R ² =2 ^2N and transforming it into a Montgomery domain, and by inverse mapping the Montgomery multiplication result, ABR. We need a calculation to make AB.

가 되어 R² 값을 계산할 수 없다. R ² values used in the Montgomery domain mapping, ^{R 2 = 2 N × 2 N} mod p is calculated by, a different number of bits of the modular value of p and p in accordance with the elliptic curve N. Since the word-based Montgomery modular multiplier 12 of the word-based finite field operator 10 according to the present invention calculates the Montgomery modular multiplication,

It becomes impossible to calculate the value of ^{R 2.}

^{In the present invention, in order to calculate the value of R 2} using the word-based Montgomery modular multiplier 12, a calculation method represented by the pseudocode of FIG. 4 is provided.

값을 효율적으로 계산할 수 있다. The method according to the pseudo-code of FIG. 4 provided by the present invention performs a simple shift and subtraction operation 2 (m×ω)-times iterative calculation without additional hardware.

You can calculate the value efficiently.

Here, ω represents the size of a word processed by the word-based finite field operator, and m=N/ω represents the number of ω-bit words constituting the modular value p.

값의 계산 과정을 살펴보도록 한다. Meanwhile, in Montgomery's modular multiplication process,

Let's look at the process of calculating the value.

가 필요하다.As shown in Fig. 2, in order to make the least significant word of the partial product addition result 0 by using congruence in the Montgomery modular multiplication operation process by the operation pseudocode of the word-based Montgomery modular multiplier, a constant value

Need

는 모듈러 값 p의 최하위 워드(ω-비트)를 나타내며, 타원곡선에 따라 모듈러 값 p와 p의 비트 수 N이 다르다. here,

Denotes the least significant word (ω-bit) of the modular value p, and the number of bits N of the modular value p and p differs according to the elliptic curve.

값을 계산하기 위해 도 5에 도시된 슈도코드로 표현되는 계산 방법을 제공한다. In the present invention, using the word-based finite field operator 10

In order to calculate the value, a calculation method represented by the pseudocode shown in FIG. 5 is provided.

의 곱의 역원

을 구한 후, 그 결과를 0에서 뺄셈 연산하여 부가적인 하드웨어 없이

값을 효율적으로 계산할 수 있다.In the method according to the pseudocode of FIG. 5 according to the present invention, the word-based Montgomery multiplication is repeatedly calculated (ω-2)-times

Inverse of the product of

After finding, the result is subtracted from zero without additional hardware.

You can calculate the value efficiently.

를 구하면, 메시지 x를 타원곡선 상의 점 (x,

)로 매핑할 수 있다. In addition, for EC-ElGamal encryption based on an elliptic curve, a process of mapping the message data x to the points (x, y) on the elliptic curve is required.To this end, the elliptic curve coefficients a and b and the elliptic curve coefficients a and b are in the elliptic curve equation of (Equation 1) below. Substituting the message data x and calculating the value of ^{y 2} _equ ^{by (Equation 2), z=y 2} _equ Modular square root of

If we get the message x, the point on the elliptic curve (x,

) Can be mapped.

이 되며, 타원곡선의 모듈러 값 p에 대해 (p+1) mod 4=0인 경우에 대해 (식 3)이 성립하므로, z의 제곱근

은 (식 4)와 같이 표현된다. ^{Since z (p-1)/2} = y ^(p-1) for z=y ² , Fermat's little theorem

Is, and (Equation 3) holds for the case of (p+1) mod 4=0 for the modular value p of the elliptic curve, so the square root of z

Is expressed as (Equation 4).

^{Therefore, z=y 2} _equ is calculated by (Equation 2), and the modular square root z _sqrt of z can be obtained by (Equation 4). However, in some cases, the modular square root of z=y ² _equ ^{does not exist, so y 2} _cal =z ² _sqrt is calculated by squaring the _{calculated z sqrt} value, and it is compared ^{with y 2} _equ calculated by (Equation 2). It is necessary to determine whether the square root is present or not. If y ² _cal =y ² _{equ , the z sqrt} value calculated by (Equation 4) is the square root of z=y ² _equ , and the message data x is the point on the elliptic curve (x, y) = (x, z _sqrt ) To complete the mapping. If y ² _cal ≠y ² _equ, then the value calculated by (Equation 4) ^{is not the square root of z=y 2} _equ . In this case, the message data is changed to x+1, and the previously described process is performed. Repeat.

Of course, new message data may be input and repeat from step 1 again.

The present invention provides a method represented by the pseudocode of FIG. 6 for mapping message data x to points (x, y) on an elliptic curve according to the process described above using the word-based finite field operator of FIG. 1.

In the pseudocode of FIG. 6 according to the present invention, A and B represent the coefficients a and b of the elliptic curve equation of (Equation 1), respectively, X represents the message data x, and p represents the modular value of the elliptic curve.

The detailed operation process according to the pseudo code of FIG. 6 is as follows.

값 계산 방법에 의해 계산된다.First, step 1 maps the message data X and the elliptic curve coefficients A and B to the Montgomery domain, respectively.

It is calculated by the value calculation method.

Step 2 calculates (p+1)/4 for the case where (p+1) mod 4=0 holds (NIST standard elliptic curves P-192, P-256, P-384, P-521). do.

Step 3 sets an initial value for calculating the square root z _sqrt ^{of y 2} _cal.

Steps 4 to 7 calculate y ² _equ by (Equation 2) by the word-based Montgomery modular multiplication method and the modular adder/subtracter of FIG. 3 according to the present invention.

_{In steps 8 to 15, z sqrt} values are calculated by the word-based Montgomery modular multiplication method of FIG. 3 according to the present invention using the relationship of (Equation 4).

In step 16, the z _sqrt value calculated in steps 8 to 15 is squared, and the y ² _cal value is calculated by the word-based Montgomery modular multiplication method of FIG. 3 according to the present invention.

If Steps 17 to 18 are y ² _cal =y ² _{equ , the z sqrt} value calculated in Steps 8 to 15 is a modular square root value, and the process moves to Step 23.

If Steps 19 to 21 are y ² _cal ≠y ² _{equ, then the z sqrt} value calculated in Steps 8 to 15 is not a modular square root value, in this case it is modular in the Montgomery domain to change the message data x to x+1. Calculate the addition (XR+R) mod p, go to step 2 and recalculate.

Alternatively, new message data may be input and repeat from step 1 again.

_{In Steps 23 and 24, since z sqrt} identified as a valid modular square root in Step-17 is a value of the Montgomery domain, x- on an elliptic curve through inverse mapping by the word-based Montgomery modular multiplication method of FIG. 3 according to the present invention. It is converted into a coordinate value X and a y-coordinate value Y.

On the other hand, the elliptic curve point calculation method includes points on the elliptic curve used in the public key cryptographic protocol by the computing device according to the present invention, scalar multiplication (PSM), addition of points (PA), doubling of points (PD), and subtraction of points ( PS) operation can be implemented, and it selectively calculates the elliptic curve point operation on the decimal body and the elliptic curve point operation on the binary body.

The Jacobian projection coordinate system is applied to the process of calculating the elliptic curve point on a prime number. You can calculate point operations with sieve subtraction. The Lopez-Dahab projection coordinate system is applied to the process of calculating the elliptic curve point on a binary body. You can calculate point operations with sieve subtraction.

A detailed operation sequence for calculating a point on an elliptic curve by the apparatus and method of the present invention is as follows.

First of all, point addition,

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(A_x, R², p)=A^* _x, WMM(A_y, R², p)=A^* _y와 WMM(B_x, R², p)=B^* _x, WMM(B_y, R², p)=B^* _y, WMM(1, R², p)=1^*에 의해 몽고메리 도메인으로 매핑 → ④ 자코비안 좌표계(소수체 상의 타원곡선의 경우) 또는 로페즈-다합 좌표계(이진체 상의 타원곡선의 경우)로 매핑된 점 A_z=(A^* _x, A^* _y, 1^*)와 B_z=(B^* _x, B^* _y, 1^*)에 대해 모듈러 가산 및 감산과 몽고메리 모듈러 곱셈의 반복 연산에 의해 점 덧셈 (Q^* _x, Q^* _y, Q^* _z)=A_z+B_z을 계산 → ⑤ 점 덧셈 연산결과 (Q^* _x, Q^* _y, Q^* _z)를 아핀 좌표계의 점 (Q^* _x/(Q^* _z)^c, Q^* _y/(Q^* _z)^d)=(P^* _x, P^* _y)으로 좌표계 변환하며, 이 때, 자코비안 좌표계의 경우에 c=2, d=3 을 적용하여 계산하고, 로페즈-다합 좌표계의 경우에는 c=1, d=2 를 적용하여 계산함 → ⑥ 아핀 좌표계로 변환된 점 (P^* _x, P^* _y)를 워드 기반 몽고메리 곱셈 WMM(P^* _x, 1, p)=P_x와 WMM(P^* _y, 1, p)=P_y에 의한 리매핑으로 점 덧셈 연산결과 (P_x, P_y)를 계산 → ⑦ 점 덧셈 연산결과 값 (P_x, P_y)를 출력」「① Enter the elliptic curve domain parameter and the points on the elliptic curve (A _x , A _y ) and (B _x , B _y ) values → ②

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(A _x , R ² , p)=A ^* _x , WMM(A _y , R ² , p)=A ^* _y and WMM(B _x , R ² , p)=B ^{Mapping to Montgomery's domain by *} _x , WMM(B _y , R ² , p)=B ^* _y , WMM(1, R ² , p)=1 ^* → ④ Jacobian coordinate system (in case of elliptic curve on a decimal point) _{Or to the point A z} =(A ^* _x , A ^* _y , 1 ^* ) and B _z =(B ^* _x , B ^* _y , 1 ^* ) mapped to the Lopez-Multisum coordinate system (for an elliptic curve on a binary body). ^{Calculate point addition (Q *} _x , Q ^* _y , Q ^* _z )=A _z +B _z → ⑤ Calculate point addition operation result (Q ^* _x , Q ^* ) by iterative operations of modular addition and subtraction and Montgomery's modular multiplication. _y , Q ^* _z ) into affine coordinate system points (Q ^* _x /(Q ^* _z ) ^c , Q ^* _y /(Q ^* _z ) ^d )=(P ^* _x , P ^* _y ) In the case of the Jacobian coordinate system, c=2, d=3 is applied, and in the case of the Lopez-multiple coordinate system, c=1, d=2 is applied → ⑥ The point converted to the affine coordinate system (P ^* _x , P ^* _y ) by word-based Montgomery multiplication WMM(P ^* _x , 1, p)=P _x and WMM(P ^* _y , 1, p)=P _y by remapping by point addition operation result (P _x , _{Calculate P y} ) → ⑦ Output the result of point addition operation (P _x , P _{y )」}

Is calculated as.

Next, double the point,

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(A_x, R², p)=A^* _x, WMM(A_y, R², p)=A^* _y와 WMM(1, R², p)=1^*에 의해 몽고메리 도메인으로 매핑 → ④ 자코비안 좌표계(소수체 상의 타원곡선의 경우) 또는 로페즈-다합 좌표계(이진체 상의 타원곡선의 경우)로 매핑된 점 A_z=(A^* _x, A^* _y, 1^*)에 대해 모듈러 가산 및 감산과 몽고메리 모듈러 곱셈의 반복 연산에 의해 점 두배 (Q^* _x, Q^* _y, Q^* _z)=2A_z를 계산 → ⑤ 점 두배 연산결과 (Q^* _x, Q^* _y, Q^* _z)를 아핀 좌표계의 점 (Q^* _x/(Q^* _z)^c, Q^* _y/(Q^* _z)^d)=(P^* _x, P^* _y)으로 좌표계 변환하며, 이 때, 자코비안 좌표계의 경우에 c=2, d=3을 적용하여 계산하고, 로페즈-다합 좌표계의 경우에는 c=1, d=2를 적용하여 계산함 → ⑥ 아핀 좌표계로 변환된 점 (P^* _x, P^* _y)를 워드 기반 몽고메리 곱셈 WMM(P^* _x, 1, p)=P_x와 WMM(P^* _y, 1, p)=P_y에 의한 리매핑으로 점 두배 연산결과 (P_x, P_y)를 계산 → ⑦ 점 두배 연산결과 값 (P_x, P_y)를 출력」「① Enter the elliptic curve domain parameter and the point (A _x , A _{y) value on the elliptic curve → ②}

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(A _x , R ² , p)=A ^* _x , WMM(A _y , R ² , p)=A ^* _y and WMM(1, R ² , p)=1 ^* Mapped to Montgomery's domain by → ④ Points mapped to the Jacobian coordinate system (for elliptic curves on decimal bodies) or Lopez-multiple coordinates (for elliptic curves on binary bodies) A _z =(A ^* _x , A ^* _y , ^{Calculate point doubling (Q *} _x , Q ^* _y , Q ^* _z ) = 2A _z by iterative operation of modular addition and subtraction and Montgomery's modular multiplication for 1 ^* ) → ⑤ Calculation of point doubling result (Q ^* _x , Q ^* _y , Q ^* _z ) into affine coordinate system points (Q ^* _x /(Q ^* _z ) ^c , Q ^* _y /(Q ^* _z ) ^d )=(P ^* _x , P ^* _y ), At this time, in the case of the Jacobian coordinate system, c=2 and d=3 are applied to the calculation, and in the case of the Lopez-multiple coordinate system, the calculation is performed by applying c=1, d=2 → ⑥ The point converted to the affine coordinate system ( ^{_{P * x, P * y)}} (P a word-based Montgomery multiplication _{^{WMM * x, 1, p)}} = P x and _{^{WMM (P * y, 1,}} p) = point doubling calculation result to remapping by P _y (P _{Calculate x} , P _y ) → ⑦ Output the point doubling operation result (P _x , P _{y )」}

Is calculated as.

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(A_x, R², p)=A^* _x, WMM(A_y, R², p)=A^* _y와 WMM(1, R², p)=1^*에 의해 몽고메리 도메인으로 매핑 → ④ 자코비안 좌표계(소수체 상의 타원곡선의 경우) 또는 로페즈-다합 좌표계(이진체 상의 타원곡선의 경우)로 매핑된 점 A_z=(A^* _x, A^* _y, 1^*)에 대해 상기 점 덧셈 연산과 상기 점 두배 연산에 의해 점 스칼라 곱셈 을 계산 (Q^* _x, Q^* _y, Q^* _z)=K×A_z을 계산 → ⑤ 점 스칼라 곱셈 연산결과 (Q^* _x, Q^* _y, Q^* _z)를 아핀 좌표계의 점 (Q^* _x/(Q^* _z)^c, Q^* _y/(Q^* _z)^d)=(P^* _x, P^* _y)으로 좌표계 변환하며, 이 때 자코비안 좌표계의 경우에 c=2, d=3을 적용하여 계산하고, 로페즈-다합 좌표계의 경우에는 c=1, d=2를 적용하여 계산함 → ⑥ 아핀 좌표계로 변환된 점 (P^* _x, P^* _y)를 워드 기반 몽고메리 곱셈 WMM(P^* _x, 1, p)=P_x와 WMM(P^* _y, 1, p)=P_y에 의한 리매핑으로 점 스칼라 곱셈 연산결과 (P_x, P_y)를 계산 → ⑦ 점 스칼라 곱셈 연산결과 값 (P_x, P_y)를 출력」Next, the point scalar multiplication is ① elliptic curve domain parameter, point on elliptic curve (A _x , A _y ) value and key value K input → ②

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(A _x , R ² , p)=A ^* _x , WMM(A _y , R ² , p)=A ^* _y and WMM(1, R ² , p)=1 ^* Mapped to Montgomery's domain by → ④ Points mapped to the Jacobian coordinate system (for elliptic curves on decimal bodies) or Lopez-multiple coordinates (for elliptic curves on binary bodies) A _z =(A ^* _x , A ^* _y , Calculate point scalar multiplication for 1 ^* ) by the point addition operation and the point doubling operation (Q ^* _x , Q ^* _y , Q ^* _z ) = K×A _z → ⑤ Point scalar multiplication result (Q ^* _x , Q ^* _y , Q ^* _z ) into points in the affine coordinate system (Q ^* _x /(Q ^* _z ) ^c , Q ^* _y /(Q ^* _z ) ^d )=(P ^* _x , P ^* _y ) The coordinate system is converted, and calculated by applying c=2 and d=3 in the case of Jacobian coordinate system, and calculating by applying c=1, d=2 in the case of Lopez-multiple coordinate system → ⑥ Converted to affine coordinate system the point ^{_{(P * x, P * y}} ) the word-based Montgomery multiplication _{^{WMM (P * x, 1,}} p) = P x and WMM point scalar multiplication as a remapping by _{^{(P * y, 1, p}} ) = P y Calculate the operation result (P _x , P _y ) → ⑦ Output the point scalar multiplication operation result value (P _x , P _{y )」}

Is calculated as.

Next, the point subtraction is,

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(A_x, R², p)=A^* _x, WMM(A_y, R², p)=A^* _y와 WMM(B_x, R², p)=B^* _x, WMM(B_y, R², p)=B^* _y, WMM(1, R², p)=1^*에 의해 몽고메리 도메인으로 매핑 → ④ 몽고메리 도메인으로 매핑된 점 B의 y-좌표 값 B^* _y에 대해 0-B^* _y mod p의 모듈러 뺄셈 연산으로 x-축에 대한 대칭점 -B^* _y를 계산 → ⑤ 자코비안 좌표계(소수체 상의 타원곡선의 경우) 또는 로페즈-다합 좌표계(이진체 상의 타원곡선의 경우)로 매핑된 점 A_z=(A^* _x, A^* _y, 1^*)와 B_z=(B^* _x,-B^* _y, 1^*)에 대해 모듈러 가산 및 감산과 몽고메리 모듈러 곱셈의 반복 연산에 의해 점 뺄셈 (Q^* _x, Q^* _y, Q^* _z)=A_z -B _z 을 계산 → ⑥ 점 뺄셈 연산결과 (Q^* _x, Q^* _y, Q^* _z)를 아핀 좌표계의 점 (Q^* _x/(Q^* _z)^c, Q^* _y/(Q^* _z)^d)=(P^* _x, P^* _y)으로 좌표계 변환 연산하며, 자코비안 좌표계의 경우에 c=2, d=3이고, 로페즈-다합 좌표계의 경우에는 c=1, d=2를 적용 → ⑦ 아핀 좌표계로 변환된 점 (P^* _x, P^* _y)를 워드 기반 몽고메리 곱셈 WMM(P^* _x, 1, p)=P_x와 WMM(P^* _y, 1, p)=P_y에 의한 리매핑으로 점 뺄셈 연산결과 (P_x, P_y)를 계산 → ⑧ 점 뺄셈 연산결과 값 (P_x, P_y)를 출력」「① Enter the elliptic curve domain parameter and the points (A _x , A _y ) and (B _x , B _{y) values on the elliptic curve → ②}

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(A _x , R ² , p)=A ^* _x , WMM(A _y , R ² , p)=A ^* _y and WMM(B _x , R ² , p)=B ^* _x , WMM(B _y , R ² , p)=B ^* _y , WMM(1, R ² , p)=1 ^* mapped to Montgomery domain → ④ y-coordinate value of point B mapped to Montgomery domain for B ^* _y modular subtraction operations of the 0-B ^* _y mod p calculate daechingjeom ^* -B _y for the x- axis → ⑤ Jacobian coordinate system (in the case of an elliptic curve on the prime number form) or Lopez-Dahab coordinate system (binary For elliptic curves on a body), the mapped points A _z = (A ^* _x , A ^* _y , 1 ^* ) and B _z = (B ^* _x , -B ^* _y , 1 ^* ), with modular addition and subtraction Subtract points by iterative operation of Montgomery's modular multiplication (Q ^* _x , Q ^* _y , Q ^* _z )=A _z -B _z → ⑥ Point subtraction operation result (Q ^* _x , Q ^* _y , Q ^* _z ) affine the point of coordinate system (Q ^* _x /(Q ^* _z ) ^c , Q ^* _y /(Q ^* _z ) ^d )= (P ^* _x , P ^* _y ) converts the coordinate system and applies c=2, d=3 in the Jacobian coordinate system, and c=1, d=2 in the Lopez-multiple coordinate system → ⑦ Affine coordinate system The converted point (P ^* _x , P ^* _y ) to the word-based Montgomery multiplication WMM(P ^* _x , 1, p)=P _x and WMM(P ^* _y , 1, p)=P _y by remapping Calculate the result of the subtraction operation (P _x , P _y ) → ⑧ Output the value of the point subtraction operation result (P _x , P _y )」

Is calculated as.

Meanwhile, the computing device 100 according to the present invention can implement modular addition (MA), modular subtraction (MS), modular multiplication (MM), modular division (MD), and modular inverse (MI) operations used in public key cryptographic protocols. I can. The detailed operation sequence for modular operation is as follows.

First of all, modular addition

① Input operands A and B and the modular value p → ② Calculate the modular addition operation Q=A+B mod p → ③ Output the modular addition operation result Q.

Next, modular subtraction

① Input operands A and B and the modular value p → ② Calculate the modular subtraction operation Q=A-B mod p → ③ Output the modular subtraction result Q.

Also modular multiplication is

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(A, R², p)=A^*에 의해 피연산자 A를 몽고메리 도메인으로 매핑 → ④ 워드 기반 몽고메리 곱셈 WMM(A^*, B, p)에 의해 모듈러 곱셈 연산 Q=A×B mod p를 계산 → ⑤ 모듈러 곱셈 연산결과 값 Q를 출력한다. ① Input operands A and B and modular value p → ②

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(A, R ² , p)=A ^* Map operand A to Montgomery domain → ④ Word-based Montgomery multiplication WMM(A ^* , B, p) by modular multiplication operation Q =A×B mod p is calculated → ⑤ Modular multiplication result value Q is output.

In addition, the modular station

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(A, R², p)=A^*와, WMM(1, R², p)=1^*에 의해 몽고메리 도메인으로 매핑 → ④ 워드 기반 몽고메리 곱셈 WMM(A^*, T, p)의 반복 연산에 의해 피연산자 A^*의 모듈러 역원

을 계산 → ⑤ 워드 기반 몽고메리 곱셈 WMM(

, 1, p)에 의한 리매핑으로 모듈러 역원 Q=A^-1 mod p 계산 → ⑥ 모듈러 역원 연산결과 값 Q를 출력한다. ① Input operand A and modular value p → ②

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(A, R ² , p)=A ^* Mapped to Montgomery domain by WMM(1, R ² , p)=1 ^* → ④ Word-based Montgomery multiplication WMM(A ^* , T, p) modular inverse of ^{operand A *}

→ ⑤ word-based Montgomery multiplication WMM(

, 1, p) ^{Calculate modular inverse Q=A -1} mod p by remapping → ⑥ Modular inverse calculation result value Q is output.

Finally, modular division

값과

값 계산 → ③ 워드 기반 몽고메리 곱셈 WMM(B, R², p)=B^*와, WMM(1, R², p)=1^*에 의해 몽고메리 도메인으로 매핑 → ④ 워드 기반 몽고메리 곱셈 WMM(B^*, T, p)의 반복 연산에 의해 피연산자 B^*의 모듈러 역원

을 계산 → ⑤ 워드 기반 몽고메리 곱셈 WMM(A,

, p)에 의해 모듈러 나눗셈 Q=(A/B) mod p를 계산 → ⑥ 모듈러 나눗셈 연산결과 값 Q를 출력한다. ① Input operands A and B and modular value p → ②

Value and

Value calculation → ③ Word-based Montgomery multiplication WMM(B, R ² , p)=B ^* and WMM(1, R ² , p)=1 ^* Mapped to Montgomery domain → ④ Word-based Montgomery multiplication WMM(B ^* , T, p) modular inverse of ^{operand B *}

→ ⑤ word-based Montgomery multiplication WMM(A,

Calculate the modular division Q=(A/B) mod p by, p) → ⑥ Output the modular division result value Q.

가 계산되는 제2단계와, 동작모드에 따른 동작모드 제어부(40)의 제어에 따라 타원곡선 점 연산(PSM, PA, PS, PD)과 유한체 모듈러 연산(MM, MD, MI, MA, MS)이 수행되는 제3단계와, 좌표계 변환 단계로, 소수체의 경우에는 자코비안(Jacobian) 좌표계에서 2차원 아핀(affine) 좌표계로, 이진체의 경우에는 로페즈-다합(Lopez-Dahab) 좌표계에서 2차원 아핀 좌표계로 변환이 수행되는 제4단계와, 몽고메리 도메인에서 계산된 연산결과에 포함되어 있는

을 제거하기 위해 리매핑(re-mapping)되는 제5단계와, 상기 제5단계의 결과데이터가 출력되는 제6단계로 이루어진다. In addition, a first step in which the elliptic curve domain parameter and data are input to the calculating device through the calculating device by the operation method according to an embodiment of the present invention, and the initial parameter according to the elliptic curve under the control of the operation mode controller 40. It is necessary to map the data to the Montgomery domain for word-based Montgomery modular multiplication as a value.

The second step in which is calculated and the elliptic curve point operation (PSM, PA, PS, PD) and finite field modular operation (MM, MD, MI, MA, MS) under the control of the operation mode controller 40 according to the operation mode. ) Is performed, in the case of a coordinate system conversion step, in the case of a minor body, from a Jacobian coordinate system to a two-dimensional affine coordinate system, and in the case of a binary body, in the Lopez-Dahab coordinate system. The fourth step in which the transformation is performed into a two-dimensional affine coordinate system, and included in the calculation result calculated in the Montgomery domain.

The fifth step is re-mapping in order to remove the data, and the sixth step is outputting the result data of the fifth step.

가 계산된다.

의 계산 과정은 전술한 바와 같다. As described above, the calculation method according to the present invention is necessary for mapping to the Montgomery domain under the control of the operation mode controller 40 after the elliptic curve domain parameter and data are input to the computing device.

Is calculated.

The calculation process of is as described above.

Then, under the control of the operation mode controller 40, elliptic curve point calculation (PSM, PA, PS, PD) and finite field modular calculation (MM, MD, MI, MA, MS) are performed, and the elliptic curve point calculation and The finite field modular calculation process is as described above.

Then, the step of transforming the coordinate system is performed. In the case of a minor body, the two-dimensional affine from the Jacobian coordinate system to a two-dimensional affine coordinate system, and in the case of a binary body, a two-dimensional affine from the Lopez-Dahab coordinate system. Transformation is performed in the coordinate system.

을 제거하기 위해 부가적인 하드웨어 없이 본 발명에 따른 연산장치를 이용하여 내부적으로 계산하는 리매핑(re-mapping) 단계가 수행되고, 최종적으로 결과가 출력되게 된다. Next, included in the calculation result calculated in the Montgomery domain

In order to remove the data, a re-mapping step that is internally calculated using the computing device according to the present invention without additional hardware is performed, and the result is finally output.

값과, 몽고메리 곱셈 과정에서 합동(congruence)을 이용한 축약 연산에 사용되는

를 계산하도록 구성되게 되며, 타원곡선 기반의 EC-ElGamal 암호화를 위해 메시지 데이터를 타원곡선 상의 점으로 매핑하는 과정에서 필요한 모듈러 제곱근 연산을 계산할 수 있어 외부의 별도 하드웨어 부가 없이 효율적이며 신속하게 계산이 이루어질 수 있다. In the calculation method according to the present invention, the computing device supports elliptic curves having multiple shapes through repetitive calculations according to the bit size of the finite field calculator, and after receiving a modular value p that varies according to the elliptic curve from the outside, Montgomery Required for modular multiplication and finite field operations

Value and the congruence used in the Montgomery multiplication process.

In the process of mapping message data to points on the elliptic curve for EC-ElGamal encryption based on an elliptic curve, the required modular square root operation can be calculated, so that the calculation can be performed efficiently and quickly without additional external hardware. I can.

As described above, the present invention has been described with reference to an embodiment shown in the drawings, but this is only exemplary, and those of ordinary skill in the art can recognize that various modifications and other equivalent embodiments are possible therefrom. I will understand.

Therefore, the true technical protection scope of the present invention should be determined by the technical spirit of the appended claims.

10: word-based finite field operator 20: memory
30: register block 40: operation mode control unit
100: operation device

Claims (7)

a word-based finite field operator 10 including a ω-bit sized modular adder/subtracter 11 and a ω-bit sized word-based Montgomery modular multiplier 12;
A memory 20 for storing an elliptic curve domain parameter and an intermediate result value of a word-based finite field operation;
A register block 30 that stores a value for setting an operation mode and stores an intermediate result value of a word-based finite field operation;
An operation mode control unit 40 that controls the word-based finite field operator 10, the memory 20, and the register block 30;
A computing device for hardware implementation of a public key cryptographic system that supports elliptic curves on prime numbers and binary bodies, comprising:
(Where ω is a natural number representing the size of a word)
The method of claim 1,
The operation mode stored in the register block 30 and controlled by the operation mode control unit 40 is
Elliptic curve point operation mode,
A computing device for hardware implementation of a public key cryptographic system that supports elliptic curves on prime numbers and binary bodies, comprising a finite field modular operation mode.
The method of claim 2,
The operation mode is
To map the data to the Montgomery domain

With the calculation mode,
An inverse mapping mode that converts the operation result of the Montgomery domain into an integer domain,
For use in operations for transforming the coordinate system and for contraction using congruence in the Montgomery modular multiplication process.

The calculation mode of the value,
A computing device for hardware implementation of a public key cryptosystem supporting an elliptic curve on a prime number and a binary body, further comprising a modular square root calculation mode required in a process of mapping an arbitrary message value to a point on an elliptic curve.
(Where R = 2 ^N , N is the number of bits representing the size of the elliptic curve, p represents a modular value for a prime number, and a short polynomial for a binary body,

Is the least significant word of the modular value p, ω is a natural number representing the size of the word)
The method of claim 1,
The word-based Montgomery modular multiplier 12 is
An ω-bit-sized dual field multiplier 12a that selectively calculates multiplication on a fractional body and multiplication on a binary body,
a dual field adder 12b of ω-bit size,
a ω-bit sized register 12c, and
1-bit carry storage register (12d),
A 2-bit carry storage register (12e),
A 1-bit adder (12f),
A computing device for hardware implementation of a public key cryptographic system that supports elliptic curves on prime numbers and binary bodies, comprising a plurality of selectors (12g).
An operation method for hardware implementation of a public key cryptographic system that supports elliptic curves on a decimal body and a binary body through the computing device according to claims 1 to 4, comprising:
A first step of inputting an elliptic curve domain parameter and data to a computing device;
As an initial parameter value according to an elliptic curve under the control of the operation mode controller 40, it is necessary to map data to the Montgomery domain for word-based Montgomery modular multiplication.

A second step in which is calculated;
The third step in which elliptic curve point calculations (PSM, PA, PS, PD) and finite field modular calculations (MM, MD, MI, MA, MS) are performed under the control of the operation mode controller 40 according to the operation mode. ;
In the case of a coordinate system conversion step, in the case of a minor body, the conversion is performed from a Jacobian coordinate system to a two-dimensional affine coordinate system, and in the case of a binary body, a two-dimensional affine coordinate system is performed. A fourth step;
Included in the calculation result calculated in the Montgomery domain

A fifth step of being remapped to remove
A sixth step of outputting the result data of the fifth step; an operation method for hardware implementation of a public key cryptographic system that supports elliptic curves on decimal and binary bodies.
(Where R = 2 ^N , N is the number of bits representing the size of the elliptic curve, p is a modular value for a prime number, and a short polynomial for a binary body)
The method of claim 5,
The operation device is
It supports elliptic curves having multiple shapes through repetitive operation according to the bit size of the finite field calculator, and after receiving the modular value p that varies according to the elliptic curve from the outside, it is necessary for Montgomery modular multiplication and finite field calculation.

Value and the congruence used in the Montgomery multiplication process.

An operation method for hardware implementation of a public key cryptographic system that supports elliptic curves on prime numbers and binary bodies, characterized in that it is configured to calculate.
The method of claim 5,
The operation device is
A public-key cryptographic system that supports elliptic curves on decimal and binary bodies, characterized in that it is configured to calculate a modular square root operation required in the process of mapping message data to points on an elliptic curve for EC-ElGamal encryption based on an elliptic curve. Method of operation for hardware implementation of

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