Apparatus and method for performing operation being secure against side channel attack

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.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority from Korean Patent Application No. 10-2017-0055699, filed on Apr. 28, 2017, the disclosure of which is incorporated herein by reference in its entirety.

BACKGROUND

The present disclosure relates to a technology for side-channel attack prevention.

2. Discussion of Related Art

As Internet of Things (IoT) devices evolve, there is a growing risk of side-channel attacks that gain important information by exploiting physical information leaked from devices during performance of mathematical computations for a key exchange, an encryption, a digital signature for encryption, etc.

A side-channel attack obtains secret information using a leakage of side-channel information (e.g., power consumption, amount of electromagnetic radiation, algorithm execution time, etc.) for computations performed during an operation of an algorithm for key exchange, encryption, digital signature, etc.

A power analysis attack, which is a form of side-channel attack, is known as the most powerful side-channel attack, and equipment for power analysis attacks is known to be a very effective attack means because of high probability of realization with low cost. Thus, the power analysis attack is a field in which a lot of research is currently being conducted. A method of such a power analysis attack largely includes simple power analysis (SPA) and differential power analysis (DPA)

For example, an exponentiation operation or a scalar multiplication operation of a point is an essential operation in a related art encryption key exchange scheme, a public key encryption scheme, a digital signature scheme, and the like. In the case of exponentiation (or scalar multiplication) operation, when an exponent value is expressed as a bit string, a square operation and a multiplication operation are performed when each bit in the bit string is 1, and only the multiplication operation is performed when the each of the bits in the bit string is 0.

The power analysis attack exploits the different power consumption requirements for the different operations performed in the case in which each bit value is 1 and in the case in which each of the bit values is 0. That is, since the power consumption for the square operation and the subsequent multiplication operation in the case in which the bit value is 1 is large and the power consumption in the case in which the bit value is 0 is small, it is possible to obtain a secret value (a secret key and the like) by taking information on an exponent value, which is the secret value, through an analysis of power consumption occurring at a time of an exponentiation computation.

Related art methods for preventing such side-channel attacks are methods for protecting against only some side-channel attacks. However, a method to secure against all of the side-channel attacks has not been proposed. In addition, related art methods for preventing side-channel attacks have a problem in that they require a large amount of computation and thus cause performance degradation, i.e., a time delay during generation of a digital signature).

SUMMARY

According to an aspect of the exemplary embodiment, there is provided an apparatus comprising: a processor configured to execute: a seed value generator configured to generate a seed value; a divider configured to divide the seed value into a plurality of blocks; a first extractor configured to extract a plurality of first parameter values from a first parameter candidate value set including a plurality of first parameter candidate values, each of the plurality of first parameter values respectively corresponding to one of the plurality of divided blocks; a second extractor configured to extract a plurality of second parameter values from a second parameter candidate value set including a plurality of second parameter candidate values generated by using each of the plurality of first parameter candidate values included in the first parameter candidate value set, each of the plurality of second parameter values respectively corresponding to one of the plurality of divided blocks; a third extractor configured to extract a plurality of third parameter values from a third parameter candidate value set including a plurality of third parameter candidate values generated by using each of the plurality of second parameter candidate values included in the second parameter candidate value set, each of the plurality of third parameter values respectively corresponding to one of the plurality of divided blocks; and a calculator configured to generate a first random number based on the plurality of first parameter values, generate a second random number based on the plurality of second parameter values, and generate a third random number based on the plurality of third parameter values, wherein the processor is further configured to encrypt data or generate a digital signature for the data based on at least one of the first random number, the second random number and the third random number.

Each of the plurality of second parameter candidate values included in the second parameter candidate value set may be generated by performing an exponentiation operation using a corresponding one of the plurality of first parameter candidate values included in the first parameter candidate value set as an exponent or by performing a scalar multiplication operation of a point using a corresponding one of the plurality of first parameter candidate values included in the first parameter candidate value set as a scalar multiplier.

The first extractor may be further configured to extract the plurality of first parameter values respectively corresponding to bit strings of the plurality of divided blocks and positions of the blocks in the seed value, from the first parameter candidate value set, the second extractor may be further configured to extract the plurality of second parameter values respectively corresponding to bit strings of the plurality of divided blocks and positions of the blocks in the seed value, from the second parameter candidate value set, and the third extractor may be further configured to extract the plurality of third parameter values respectively corresponding to bit strings of the plurality of divided blocks and positions of the blocks in the seed value, from the third parameter candidate value set.

The calculator may generate the first random number by adding the plurality of first parameter values to each other.

The second random number may equal to a value obtainable by performing an exponentiation operation using the first random number as an exponent or by performing a scalar multiplication operation of a point using the first random number as a scalar multiplier and using the plurality of second parameter values.

At least one of the plurality of third parameter candidate values included in the third parameter candidate value set may be obtained by multiplying at least one of the second parameter candidate values included in the second parameter candidate value set by a secret key.

The third random number may equal to a product of the second random number and the secret key.

The processor may be further configured to encrypt the data or generate the digital signature for the data based on the first random number, the second random number, and the third random number.

The seed value may comprise a random bit string.

According to another aspect of the exemplary embodiment, there is provided a method, comprising: generating a seed value; dividing the seed value into a plurality of blocks; extracting a plurality of first parameter values from a first parameter candidate value set including a plurality of first parameter candidate values, each of the plurality of first parameter values respectively corresponding to one of the plurality of divided blocks; extracting a plurality of second parameter values from a second parameter candidate value set including a plurality of second parameter candidate values generated by using each of the plurality of first parameter candidate values included in the first parameter candidate value set, each of the plurality of second parameter values respectively corresponding to one of the plurality of divided blocks; extracting a plurality of third parameter values from a third parameter candidate value set including a plurality of third parameter candidate values generated by using each of the plurality of second parameter candidate values included in the second parameter candidate value set, each of the plurality of third parameter values respectively corresponding to one of the plurality of divided blocks; generating a first random number based on the plurality of first parameter values; generating a second random number based on the plurality of second parameter values; generating a third random number based on the plurality of third parameter values; and applying at least one of the first random number, the second random number and the third random number to encrypt data or generate a digital signature for the data.

Each of the plurality of second parameter candidate values included in the second parameter candidate value set is generated by performing an exponentiation operation using a corresponding one of the plurality of first parameter candidate values included in the first parameter candidate value set as an exponent or by performing a scalar multiplication operation of a point using a corresponding one of the plurality of first parameter candidate values included in the first parameter candidate value set as a scalar multiplier.

The generating of the first random number may generate the first random number by adding the plurality of first parameter values to each other.

The second random number may be equal to a value obtainable by performing an exponentiation operation using the first random number as an exponent or by performing a scalar multiplication operation of a point using the first random number as a scalar multiplier and using the plurality of second parameter values.

At least one of the plurality of third parameter candidate values included in the third parameter candidate value set may be values obtained by multiplying at least one of the second parameter candidate values included in the second parameter candidate value set by a secret key.

The third random number may be equal to a product of the second random number and the secret key.

The encrypting the data or the generating the digital signature may be based on the first random number, the second random number, and the third random number.

The seed value may comprise a random bit string.

According to another aspect of an exemplary embodiment, there is provided a non-transitory computer readable medium having stored thereon a program for causing a computer to execute a method, comprising: generating a seed value; dividing the seed value into a plurality of blocks; extracting a plurality of first parameter values from a first parameter candidate value set including a plurality of first parameter candidate values, each of the plurality of first parameter values respectively corresponding to one of the plurality of divided blocks; extracting a plurality of second parameter values from a second parameter candidate value set including a plurality of second parameter candidate values generated by using each of the plurality of first parameter candidate values included in the first parameter candidate value set, each of the plurality of second parameter values respectively corresponding to one of the plurality of divided blocks; extracting a plurality of third parameter values from a third parameter candidate value set including a plurality of third parameter candidate values generated by using each of the plurality of second parameter candidate values included in the second parameter candidate value set, each of the plurality of third parameter values respectively corresponding to one of the plurality of divided blocks; generating a first random number based on the plurality of first parameter values; generating a second random number based on the plurality of second parameter values; generating a third random number based on the plurality of third parameter values; and applying at least one of the first random number, the second random number and the third random number to encrypt data or generate a digital signature for the data.

According to another aspect of an exemplary embodiment, there is provided a an apparatus comprising: a processor configured to execute: receive a seed value; divide the seed value into a plurality of blocks; extract, for each of the plurality of blocks, a first parameter value from a first parameter candidate value set comprising a plurality of first parameter candidate values, by using a number of the respective block and a value in the respective block as a first index; extract, for each of the plurality of blocks, a second parameter value from a second parameter candidate value set comprising a plurality of second parameter candidate values generated by using the plurality of first parameter candidate values, by using a number of the respective block and a value in the respective block as a second index; extract, for each of the plurality of blocks, a third parameter value from a third parameter candidate value set comprising a plurality of third parameter candidate values generated by using the plurality of second parameter candidate values, by using a number of the respective block and a value in the respective block as a first third; generate a first random number based on the first parameter values for each of the plurality of blocks; generate a second random number based on the second parameter values for each of the plurality of blocks; generate a third random number based on the third parameter values for each of the plurality of blocks; and apply at least one of the first random number, the second random number and the third random number to encrypt data or generate a digital signature for the data.

At least one of a number of the first parameter candidate values included in the first parameter candidate value set, a number of the second parameter candidate values included in the second parameter candidate value set, and a number of the third parameter candidate values included in the third parameter candidate value set, may change according to a length of the seed value and a total number of the plurality of blocks.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, detailed exemplary embodiments of the present disclosure will be described with reference to the accompanying drawings. The following detailed description is provided for a more comprehensive understanding of methods, devices, and/or systems described in this specification. However, the methods, devices, and/or systems are only examples, and the present disclosure is not limited thereto.

In the description of the present disclosure, detailed descriptions of related well-known functions that are determined to unnecessarily obscure the gist of the present disclosure will be omitted. Some terms described below are defined in consideration of functions thereof in the present disclosure, and meanings thereof may vary depending on, for example, a user or operator's intention or custom. Therefore, the meanings of terms should be interpreted on the basis of the scope throughout this specification. The terminology used in the detailed description is provided only to describe exemplary embodiments of the present disclosure and not for purposes of limitation. Unless the context clearly indicates otherwise, the singular forms include the plural forms. It should be understood that the terms “comprises” or “includes,” when used herein. specify the presence of some features, numbers, steps, operations, elements, and/or combinations thereof, but do not preclude the presence or possibility of addition of one or more other features, numbers, steps, operations, elements, and/or combinations thereof.

FIG. 1is a configuration diagram illustrating a computation apparatus according to an exemplary embodiment of the present disclosure.

Referring toFIG. 1, a computation apparatus100according to ab exemplary embodiment of the present disclosure includes a seed value generator110, a divider120, an extractor130, and a calculator140.

The seed value generator110generates a seed value. According to an exemplary embodiment, the seed value may be formed by a random bit string.

In this case, the seed value generator110may generate the seed value, for example, by sequentially generating arbitrary bit values.

In another example, the seed value generator110may generate the seed value by converting an ID obtained from an external device into an arbitrary bit string. In this case, a hash function, for example, SHA-256, may be used to convert the obtained ID into the arbitrary bit string.

However, a seed value generation method of the seed value generator110is not particularly limited, and the seed value generator110may generate the seed value using various known methods capable of generating a bit string of a predetermined length.

The divider120divides the seed value generated by the seed value generator110into a plurality of blocks.

Specifically, according to an exemplary embodiment of the present disclosure, the divider120may divide the seed value by a predetermined size to generate the plurality of blocks. In this case, the number of blocks generated by the divider120may be changed according to an exemplary embodiment.

For example, when the seed value generated by the seed value generator110is a bit string with a length of 256 bits, the divider120may divide the seed value into units of 8 bits to generate 32 blocks.

The extractor130extracts a plurality of second parameter values that respectively correspond to the plurality of blocks, which are generated by the divider120, from a second parameter candidate value set including second parameter candidate values generated by using each of a plurality of first parameter candidate values.

In this case, according to an exemplary embodiment of the present disclosure, the plurality of first parameter candidate values may be random values generated in advance and stored in a storage.

In addition, according to an exemplary embodiment of the present disclosure, the second parameter candidate values included in the second parameter candidate value set may be values obtained in advance through an exponentiation operation using each of the plurality of first parameter candidate values as an exponent or through a scalar multiplication operation of a point using each of the plurality of first parameter candidate values as a scalar multiplier.

According to an exemplary embodiment of the present disclosure, the second parameter candidate value set may include 2n×m second parameter candidate values. Here, m represents the number of blocks generated by the divider120, and n represents a length of a bit string included in each of the generated blocks. That is, the number of the second parameter candidate values included in the second parameter candidate value set may be changed according to the length of the seed value generated by the seed value generator110and the number of blocks generated by the divider120.

According to an exemplary embodiment of the present disclosure, the extractor130may extract second parameter values respectively corresponding bit strings in the plurality of blocks generated by the divider120and positions of the blocks in the seed value, from the second parameter candidate value set.

Specifically, each second parameter candidate value included in the second parameter candidate value set may be indexed to one of 2ndifferent bit strings, each having a length of n bits, and a position of a block including the one of 2ndifferent bit strings in the seed value. Accordingly, the extractor130may compare the bit string included in each of the blocks generated by the divider120and the position of each of the blocks in the seed value with an index value of each of the second parameter candidate values included in the second parameter candidate value set to extract a corresponding second parameter value as a second parameter value.

FIG. 2is a table showing one example of the second parameter candidate value set according to an exemplary embodiment of the present disclosure.

In the example shown inFIG. 2, a seed value generated by the seed value generator110has a length of 256 bits and that the divider120divides the seed value into 32 blocks, each having a length of 8 bits. Thus, the second parameter candidate value set210shown inFIG. 2includes 28×32 second parameter candidate values.

Specifically, a second parameter candidate value ri,jmay be generated through, for example, an exponentiation operation using a first parameter candidate value ki,jas an exponent, as shown in the following Equation 1.
ri,j=gki,jmodp[Equation 1]

In Equation 1, p represents an arbitrary prime number and g represents a generator of a multiplicative group having p as an order.

In another example, the second parameter candidate value ri,jmay be generated by performing a scalar multiplication operation of a point using the first parameter candidate value ki,jas a scalar multiplier as shown in Equation 2 and Equation 3.
ki,j·P=(xi,j,yi,j)  [Equation 2]
ri,j=xi,jmodp[Equation 3]

In Equations 2 and 3, p represents an arbitrary prime number and P represents a generator of an additive group having p as an order.

Meanwhile, each of the second parameter candidate values included in a second parameter candidate value set210corresponds to one of 256 bit strings, each having a length of 8 bits, and a position of a block including the corresponding bit string in the seed value.

Specifically, r1,1is a second parameter candidate value indexed to a bit string of 00000000 and the first block (i.e., 1 block) in the seed value, r256,32is a second parameter candidate value indexed to a bit string of 11111111 and the 32ndblock (i.e., 32 block) in the seed value.

FIG. 3is a diagram illustrating an example of an extraction of a second parameter value according to an exemplary embodiment of the present disclosure.

In the example shown inFIG. 3, a seed value310generated by the seed value generator110has a length of 256 bits and that the divider120divides the seed value310into 32 blocks, each having a length of 8 bits. In addition, a second parameter candidate value set330is the same as the second parameter candidate value set210shown inFIG. 2. According to another exemplary embodiment, the second parameter candidate value set330may be different from the second parameter candidate value set210shown inFIG. 2.

In the example shown inFIG. 3, since a bit string included in the first block of the 32 blocks generated by dividing the seed value310is “1111110,” the extractor130may extract a second parameter candidate value r255,1corresponding to “1 block” and the bit string of “1111110” from the second parameter candidate value set330as a second parameter value corresponding to the first block.

In addition, since a bit string included in the second block of the blocks generated by dividing the seed value310is “00000010,” the extractor130may extract a second parameter candidate value r3,2corresponding to “2 block” and the bit string of “00000010” from the second parameter candidate value set330as a second parameter value corresponding to the second block.

In the same way, the extractor130may extract second parameter values that respectively correspond to the 32 blocks generated by dividing the seed value310from the second parameter candidate value set330.

Referring back toFIG. 1, the calculator140generates a random number using the second parameter values extracted by the extractor130.

In this case, according to an exemplary embodiment of the present disclosure, the calculator140may generate the random number by multiplying or adding the second parameter values extracted from the second parameter candidate value set.

When the foregoing Equation 1 is applied to Equation 4, Equation 4 may be expressed as the following Equation 5.
r=g(k255,1+k3,2+k1,3+ . . . +k3,30+k256,32+k255,32)modp=gkmodp[Equation 5]

That is, the random number r generated by Equation 4 is the same as a value obtained by performing an exponentiation operation which uses k, which is the sum of the first parameter candidate values of {k255,1, k3,2, k1,3, . . . , k3,30, k256,31, k255,32} that are used as exponents to generate the second parameter values extracted by the extractor130.

When the foregoing Equation 3 is applied to Equation 6, Equation 6 may be expressed as the following Equation 7.
r=(x255,1+x3,2+x1,3+ . . . +x3,30+x256,31+x255,32)modp[Equation 7]

In addition, it can be seen from Equation 2 that (x255,1+x3,2+x1,3+ . . . +x3,30+x256,31+x255,32) in Equation 7 is the same as a value obtained using a scalar multiplication operation of a point shown in the following Equation 8.
(k255,1+k3,2+k1,3+ . . . +k3,30+k256,31+k255,32)·P=k·P[Equation 8]

Accordingly, the random number r generated by Equation 6 is the same as the value obtained by performing the scalar multiplication operation of a point which uses k, which is the sum of the first parameter candidate values of {k255,1, k3,2, k1,3, . . . , k3,30, k256,31, k255,32} that are used as scalar multipliers to generate the second parameter values extracted by the extractor130.

According to another exemplary embodiment, the calculator140may generate the random number r using various forms of one-way function other than the addition or multiplication of the second parameter values, as shown in the foregoing Equation 4 and Equation 6.

Furthermore, according to an exemplary embodiment of the present disclosure, the calculator140may encrypt data or generate a digital signature using the generated random number r.

Specifically, the calculator140may encrypt data or generate a digital signature using various types of encryption algorithm or digital signature generation algorithm which require an exponentiation operation or a scalar multiplication operation of a point for encryption or digital signature generation. In this case, the calculator140may not directly perform the exponentiation operation or the scalar multiplication operation of a point. But instead, the calculator140may use the random number r as a resulting value of the exponentiation operation or the scalar multiplication operation of a point. In this manner, according to an exemplary embodiment, since a direct exponentiation operation or a direct scalar multiplication operation of a point is not performed for a key exchange, an encryption, or a digital signature for encryption, exploitation of physical information leaked from devices by side-channel attacks during performance of mathematical computations is prevented.

For example, in the case of an identification (ID) based public key encryption, a public key of a recipient may be generated from an ID of a recipient of encrypted data, and the encrypted data which is encrypted using the generated public key is transmitted to the recipient. In this case, the seed value generator110may generate a seed value by converting the ID of the recipient into a random bit string, and the divider120may divide the generated seed value into a plurality of blocks. Thereafter, the extractor130may extract second parameter values that correspond to the respective generated blocks, from the second parameter candidate value set, and the calculator140may generate the random number r using the second parameter values and use the generated random number r as the public key of the recipient. In this case, the second parameter candidate value set may be generated in advance by an external device, such as a key generation server, and provided by performing a secure channel.

As a specific example, the calculator140may generate the following ciphertext C1 by selecting an arbitrary random number t.
C1=gt(modp)

Then, the calculator140may generate the following ciphertext C2 using the random number r, which is generated using the second parameter values extracted from the second parameter candidate value set, as a public key pk.
C2=(pktmodp)XORM=((gx)tmodp)XORM

Here, M represents a message to be sent and x represents a secret key of the recipient to which the data is transmitted.

Meanwhile, in an exemplary embodiment, the seed value generator110, the divider120, the extractor130, and the calculator140, which are shown inFIG. 1, may be implemented on one or more computing devices including one or more processors and a computer-readable recording medium connected to the one or more processors. The computer-readable recording medium may be provided inside or outside the processor, and may be connected to the processor by various well-known means. The processor in the computing device may enable the computing device to operate according to the exemplary embodiments described in the specification. For example, the processor may execute instructions stored in the computer readable recording medium, and the instructions stored in the computer readable recording medium may cause a series of operational steps to be performed on the computing device according to the exemplary embodiments of the present disclosure described in the specification when executed by the processor.

FIG. 4is a configuration diagram illustrating a computation apparatus according to another exemplary embodiment of the present disclosure.

Referring toFIG. 4, a computation apparatus400according to another exemplary embodiment of the present disclosure includes a seed value generator410, a divider420, a first extractor430, a second extractor440, and a calculator450.

In the example shown inFIG. 4, the seed value generator410, the divider420, and the second extractor440have the same configurations as those of the seed value generator110, the divider120, and the extractor130ofFIG. 1, respectively, and thus detailed descriptions thereof will not be reiterated.

The first extractor430extracts a plurality of first parameter values that respectively correspond to a plurality of blocks generated by the divider420from a first parameter candidate value set including a plurality of first parameter candidate values. According to an exemplary embodiment, the plurality of first parameter candidate values may be the same values used to generate each of the second parameter candidate values included in the above-described second parameter candidate value set inFIGS. 2 and 3.

In this case, according to an exemplary embodiment of the present disclosure, the first parameter candidate value set may include 2n×m first parameter candidate values. Moreover, the number of the first parameter candidate values included in the first parameter candidate value set may be changed according to a length of the seed value generated by the seed value generator410and the number of blocks generated by the divider420.

According to an exemplary embodiment, the first extractor430may extract first parameter values respectively corresponding to bit strings in the plurality of blocks generated by the divider420and positions of the blocks in the seed value from the first parameter candidate value set.

Specifically, each first parameter candidate value included in the first parameter candidate value set may be indexed to one of 2ndifferent bit strings, each having a length of n bits, and a position of a block including the one of 2ndifferent bit strings in the seed value.

Accordingly, the first extractor430may compare the bit string included in each of the blocks generated by the divider420and the position of each of the blocks in the seed value with an index value of each of the first parameter candidate values included in the first parameter candidate value set to extract the corresponding first parameter candidate value as a first parameter value.

Meanwhile, each of the first parameter candidate values included in the first parameter candidate value set may have the same index value as that of a corresponding second parameter candidate value included in the second parameter candidate value set.

FIG. 5is a table showing an example of the first parameter candidate value set according to an exemplary embodiment of the present disclosure.

FIG. 6is a diagram illustrating one example of an extraction of the first parameter value according to an exemplary embodiment of the present disclosure.

In the example shown inFIG. 6, a seed value610is the same as the seed value310shown inFIG. 3and that the divider420divides the seed value610into 32 blocks, each having a length of 8 bits. In addition, a first parameter candidate value set630is the same as the first parameter candidate value set510shown inFIG. 1. However, according to another exemplary embodiment, the first parameter candidate value set630may be different from the first parameter candidate value set510shown inFIG. 1.

In the example shown inFIG. 6, when a bit string included in the first block of the 32 blocks generated by dividing the seed value610is “1111110,” the first extractor430may extract a first parameter candidate value k255,1corresponding to “1 block” and the bit string of “1111110” from the first parameter candidate value set630as a first parameter value corresponding to the first block.

In addition, when a bit string included in the second block of the blocks generated by dividing the seed value610is “00000010,” the first extractor430may extract a first parameter candidate value k3,2corresponding to “2 block” and the bit string of “00000010” from the first parameter candidate value set630as a first parameter value corresponding to the second block.

In the same way, the first extractor430may extract parameter values that respectively correspond to the 32 blocks generated by dividing the seed value610from the first parameter candidate value set630.

Referring back toFIG. 4, the calculator450generates a first random number using the first parameter values extracted by the first extractor430, and generates a second random number using the second parameter values extracted by the second extractor440.

In this case, according to an exemplary embodiment of the present disclosure, the calculator450may generate a first random number by adding the first parameter values extracted from the first parameter candidate value set to each other.

For example, the calculator450may generate a first random number k from the first parameter values extracted in the example shown inFIG. 6using the following Equation 9.
k=k255,1+k3,2+k1,3+ . . . +k3,30+k256,31+k255,32[Equation 9]

Meanwhile, the calculator450may generate the first random number k using various types of one-way function in addition to the foregoing Equation 9.

Meanwhile, according to an exemplary embodiment of the preset disclosure, the calculator450may encrypt data or generate a digital signature using the generated second random number r.

Specifically, the calculator450may encrypt data or generate a digital signature using various types of encryption algorithm or digital signature generation algorithm which require an exponentiation operation or a scalar multiplication operation of a point for encryption or digital signature generation. In this case, the calculator450may not directly perform the exponentiation operation or the scalar multiplication operation of a point but may use the second random number r as a resulting value of the exponentiation operation or the scalar multiplication operation of a point.

In this manner, according to an exemplary embodiment, since a direct exponentiation operation or a direct scalar multiplication operation of a point is not performed for a key exchange, an encryption, or a digital signature for encryption, exploitation of physical information leaked from devices by side-channel attacks during performance of mathematical computations is prevented.

For example, the calculator450may generate a digital signature using a digital signature algorithm (DSA), which is one digital signature scheme.

Specifically, a digital signature according to a DSA is generated as follows:

1) The random integer k is selected (k∈[1,q−1])

2) r=(gkmod p) mod q is computed (where p is an arbitrary prime number, q is a prime divisor of p−1, and g is a generator of a multiplicative group having p as an order)

3) s=k−1(H(m)+cr) mod q is computed (where c is a secret key, m is a message, and H( ) is a hash function)

4) A signature value (r,s) is output for a message.

In this case, the calculator450may use the first random number as the random integer k and use the second random number as the signature value r. That is, the signature value r may be obtained by performing an operation, such as a multiplication operation, which is secure against side-channel attack and uses the second parameter values extracted from the second parameter candidate value set, rather than by performing an exponentiation operation using the random integer k. Accordingly, the random integer k used to generate the signature value r cannot be obtained by performing a side-channel attack, and the secret key c used for generating the signature value s is also secured.

In another example, the calculator450may generate a digital signature using an elliptic curve digital signature algorithm (ECDSA), which is one digital signature scheme.

Specifically, a digital signature according to the ECDSA is generated as follows:

1) The random integer k (k∈[1,q−1]) is selected (where q is a prime divisor of p−1 and p is an arbitrary prime number)2) k·P=(x, y) is computed (where P is a generator of an additive group having p as an order)3) r=x mod p is computed4) s=k−1(H(m)+cr) mod (p−1) is computed (where c is a secret key, m is a message, and H( ) is a hash function)

5) The signature value (r,s) is output for a message

In this case, the calculator450may use the first random number as the random integer k and use the second random number as the signature value r. That is, the signature value r may be obtained by performing an operation, such as an addition operation, which is secure against side-channel attack and uses the second parameter values extracted from the second parameter candidate value set, rather than by performing a scalar multiplication operation of a point using the random integer k. Accordingly, the random integer k used to generate the signature value r cannot be obtained by performing a side-channel attack, and the secret key c used for generating the signature value s is also secured.

In an exemplary embodiment, the seed value generator410, the divider420, the first extractor430, the second extractor440, and the calculator450, which are shown inFIG. 4, may be implemented on one or more computing devices including one or more processors and a computer-readable recording medium connected to the one or more processors. The computer-readable recording medium may be provided inside or outside the processor, and may be connected to the processor by various well-known means. The processor in the computing device may enable the computing device to operate according to the exemplary embodiments described in the specification. For example, the processor may execute instructions stored in the computer readable recording medium, and the instructions stored in the computer readable recording medium may cause a series of operational steps to be performed on the computing device according to the exemplary embodiments of the present disclosure described in the specification when executed by the processor.

FIG. 7is a configuration diagram illustrating a computation apparatus according to another embodiment of the present disclosure.

Referring toFIG. 7, a computation apparatus700according to an exemplary embodiment of the present disclosure includes a seed value generator710, a divider720, a first extractor730, a second extractor740, a third extractor750, and a calculator760.

In the example shown inFIG. 7, the seed value generator710, the divider720, the first extractor730, and the second extractor740have the same configurations as those of the seed value generator410, the divider420, the first extractor430, and the second extractor440, respectively, and thus detailed descriptions thereof will not be reiterated.

The third extractor750extracts a plurality of third parameter values that respectively correspond to a plurality of blocks generated by the divider720from a third parameter candidate value set including a plurality of third parameter candidate values generated using a plurality of second parameter candidate values included in a second parameter candidate value set.

According to an exemplary embodiment of the present disclosure, the third parameter candidate value set may include 2n×m third parameter candidate values, like in the first parameter candidate value set and the second parameter candidate value set. That is, the number of the third parameter candidate values included in the third parameter candidate value set may be changed according to a length of a seed value generated by the seed value generator710and the number of blocks generated by the divider720.

In addition, according to an exemplary embodiment of the present disclosure, the third parameter candidate values included in the third parameter candidate value set are generated using the second parameter candidate values included in the second parameter candidate value set, and at least some of the third parameter candidate values included in the third parameter candidate value set may include a value multiplied by a secret key used for encryption or digital signature generation.

Specifically, the third parameter candidate values included in the third parameter candidate value set may be, for example, values obtained by multiplying each of the second parameter candidate values included in the second parameter candidate value set by the secret key.

In another example, some of the third parameter candidate values included in the third parameter candidate value set may be identical to some of the second parameter candidate values included in the second parameter candidate value set and the remaining third parameter candidate values in the third parameter candidate value set may be values obtained by multiplying each of the remaining second parameter candidate values in the second parameter candidate value set by the secret key.

According to an exemplary embodiment of the present disclosure, the third parameter candidate values included in the third parameter candidate value set may be indexed to one of 2ndifferent bit strings, each having a length of n bits, and a position of a block including the one of 2ndifferent bit strings in the seed value. Accordingly, the third extractor750may compare the bit string included in each of the blocks generated by the divider720and the position of each of blocks in the seed value with an index value of each of the third parameter candidate values included in the third parameter candidate value set to extract the corresponding third parameter candidate value as a third parameter value.

Meanwhile, each of the third parameter candidate values included in the third parameter candidate value set may have the same index value as that of a corresponding second parameter candidate value included in the second parameter candidate value set.

FIG. 8is a table showing one example of the third parameter candidate value set according to an exemplary embodiment of the present disclosure.

In the example shown inFIG. 8, a third parameter candidate value set810includes the same number (i.e., 28×32) of third parameter candidate values as the number of second parameter candidate values included in the second parameter candidate value set210shown inFIG. 2.

Meanwhile, except for third parameter candidate values (i.e., cr1,32, cr2,32, cr3,32, . . . , cr255,32, cr256,32) having “32 block” as index values in the third parameter candidate value set810, the remaining third parameter candidate values have values equal to the second parameter candidate values of the second parameter candidate value set210that have the same index values as the third parameter candidate values.

Conversely, the third parameter candidate values (i.e., cr1,32, cr2,32, cr3,32, cr255,32, cr256,32) having “32 block” as index values in the third parameter candidate value set810have values equal to values obtained by multiplying each of the second parameter candidate values (i.e., r1,32, r2,32, r3,32, . . . , r255,32, r256,32) in the second parameter candidate value set210that have the same index values as the third parameter candidate values by a secret key c.

Specifically, a third parameter candidate value r1,1included in the third parameter candidate value set810is the same as the second parameter candidate value r1,1included in the second parameter candidate value set210and is indexed to a bit string of “00000000” and the first block (i.e., 1 block).

In addition, a third parameter candidate value cr1,32is a third parameter candidate value generated by multiplying a second parameter candidate value r1,32in the second parameter candidate value set210by the secret key c and is indexed to the bit string of “00000000” and the 32ndblock (i.e., 32 block) in the same way as the second parameter candidate value r1,32.

Meanwhile, in the example shown inFIG. 8, the third parameter candidate values (i.e., cr1,32, cr2,32, cr3,32, . . . , cr255,32, cr256,32) having “32 block” as index values are shown as having values generated by multiplying each of the second parameter candidate values (i.e., r1,32, r2,32, r3,32, . . . , r255,32, r256,32) having the same index values as those of the third parameter candidate values by the secret key c, but the third parameter candidate values are not limited thereto. Specifically, third parameter candidate values included in a third parameter candidate value set which have a specific block as index values may have values equal to values obtained by multiplying each of the second parameter candidate values having the identical block by the secret key c as index values.

FIG. 9is a table showing another example of the third parameter candidate value set according to an exemplary embodiment of the present disclosure.

Specifically,FIG. 9shows an example of the third parameter candidate value set including third parameter candidate values generated from the second parameter candidate values contained in the second parameter candidate value set210shown inFIG. 2.

In the example shown inFIG. 9, a third parameter candidate value set910includes the same number (i.e., 28×32) of third parameter candidate values as the number of second parameter candidate values in the second parameter candidate value set210shown inFIG. 2.

In addition, each of the third parameter candidate values included in the third parameter candidate value set910has a value equal to a value obtained by multiplying the corresponding second parameter candidate value having the same index value as that of the third parameter candidate value in the second parameter candidate value set210by the secret key c.

In addition, a third parameter candidate value cr255,32is a third parameter candidate value generated by multiplying a second parameter candidate value r255,32in the second parameter candidate value set210by the secret key c and is indexed to the bit string of “11111110” and the 32ndblock (i.e., 32 block) in the same way as the second parameter candidate value r255,32.

FIG. 10is a diagram illustrating one example of an extraction of the third parameter value according to an exemplary embodiment of the present disclosure.

In the example shown inFIG. 10, a seed value1010is the same as the seed values310and610shown inFIGS. 3 and 6and that the divider720divides the seed value1010into 32 blocks, each having a length of 8 bits. In addition, a third parameter candidate value set1030is the same as the third parameter candidate value set810shown inFIG. 8.

In the example shown inFIG. 8, since a bit string included in the first block of the 32 blocks generated by dividing the seed value1010is “1111110,” the third extractor750may extract a parameter candidate value r255,1that corresponds to “1 block” and the bit string of “1111110” from the third parameter candidate value set1030as a third parameter value corresponding to the first block.

In addition, since a bit string included in the second block of the blocks generated by dividing the seed value1010is “00000010,” the third extractor750may extract a third parameter candidate value r3,2that corresponds to “2 block” and the bit string of “00000010” from the third parameter candidate value set1030as a third parameter value corresponding to the second block.

In the same way, the third extractor750may extract parameter values that respectively correspond to the 32 blocks generated by dividing the seed value1010from the third parameter candidate value set1030.

FIG. 11is a diagram illustrating another example of the extraction of the third parameter value according to an exemplary embodiment of the present disclosure.

In the example shown inFIG. 11, it is assumed that a seed value1110is the same as the seed values310and610shown inFIGS. 3 and 6and that the divider720divides the seed value1110into 32 blocks, each having a length of 8 bits. In addition, it is assumed that a third parameter candidate value set1130is the same as the third parameter candidate value set910shown inFIG. 9.

In the example shown inFIG. 11, since a bit string included in the first block of the 32 blocks generated by dividing the seed value1110is “1111110,” the third extractor750may extract a parameter candidate value cr255,1that corresponds to “1 block” and the bit string of “1111110” from the third parameter candidate value set1130as a third parameter value corresponding to the first block.

In addition, since a bit string included in the second block of the blocks generated by dividing the seed value1110is “00000010,” the third extractor750may extract a third parameter candidate value cr3,2that corresponds to “2 block” and the bit string of “00000010” from the third parameter candidate value set1130as a third parameter value corresponding to the second block.

In the same way, the third extractor750may extract third parameter values that respectively correspond to the 32 blocks generated by dividing the seed value1110from the third parameter candidate value set1130.

Referring back toFIG. 7, the calculator760generates a first random number using the first parameter values extracted by the first extractor730, generates a second random number using the second parameter values extracted by the second extractor740, and generates a third random number using the third parameter values extracted by the third extractor750.

In this case, the generation of the first and second random numbers is described above, and thus redundant description will be omitted.

According to an exemplary embodiment of the present disclosure, the calculator760may generate the third random number by multiplying or adding the third parameter values extracted from the third parameter candidate value set with each other.

For example, the calculator760may use the following Equation 10 to generate a third random number cr from the third parameter values extracted in the example shown inFIG. 10.
cr=r255,1×r3,2×r1,3× . . . ×r3,30×r256,31×cr255,32[Equation 10]

In another example, the calculator760may use the following Equation 11 to generate the third random number cr from the third parameter values extracted in the example shown inFIG. 11.
cr=cr255,1+cr3,2+cr1,3+ . . . +cr3,30+cr256,31+cr255,32[Equation 11]

That is, the third random number cr generated by the calculator760is equal to the secret key c multiplied by the second random number r.

According to an exemplary embodiment, the calculator760may generate the third random number cr from the third parameter values using various forms of one-way function capable of generating a value equal to the second random number r multiplied by the secret key c, other than the addition or multiplication of the third parameter values.

Meanwhile, according to an exemplary embodiment of the present disclosure, one or more processors of the computing device100may encrypt data or generate a digital signature using the generated first random number k, second random number r, and third random number cr. According to exemplary embodiment, the calculator760may encrypt data or generate a digital signature using the generated first random number k, the second random number r, and the third random number cr. According to another exemplary embodiment, another component or another element of the one or more processors of the computing device100may encrypt data or generate a digital signature using the generated first random number k, second random number r, and third random number cr.

Specifically, the calculator760may encrypt data or generate a digital signature using various types of encryption algorithm or digital signature generation algorithm which require an exponentiation operation or a scalar multiplication operation of a point, and an operation of multiplying a value generated by performing the exponentiation operation or the scalar multiplication operation of a point by a secret key for encryption or digital signature generation. In this case, the calculator760may not directly perform the exponentiation operation or the scalar multiplication operation of a point, but may use the second random number r as a resulting value of the exponentiation operation or the scalar multiplication operation of a point. In addition, the calculator760may use the third random number cr as a resulting value of the multiplication of the second random number r and the secret key c without directly multiplying the second random number r by the secret key c. In this manner, according to an exemplary embodiment, since a direct exponentiation operation or a direct scalar multiplication operation of a point is not performed for a key exchange, an encryption, or a digital signature for encryption, exploitation of physical information leaked from devices by side-channel attacks during performance of mathematical computations is prevented.

For example, in the case in which the digital signature is generated using the DSA, as described above, the calculator760may use the first random number as the random integer k and use the second random number as the signature value r. In addition, the calculator760may generate the signature value s using the third random number obtained by performing another operation, such as an addition operation, which is secure against side-channel attack and uses the first random number and the third parameter values extracted from the third parameter candidate value set. That is, it is possible to generate the signature value r without performing an exponentiation operation using the random integer k and it is possible to generate the signature value s without performing an operation of multiplying the signature value r by the secret key c. Therefore, the random integer k used to generate the signature value r and the secret key c used to generate the signature value s cannot be obtained by performing a side-channel attack.

In another example, in the case in which the digital signature is generated using the ECDSA, as described above, the calculator760may use the first random number as the random integer k and use the second random number as the signature value r. In addition, the calculator760may generate the signature value s using the third random number obtained by performing another operation, such as a multiplication operation, which is secure against side-channel attack and uses the first random number and the third parameter values extracted from the third parameter candidate value set. That is, it is possible to generate the signature value r without performing a scalar multiplication operation of a point using the random integer k and it is possible to generate the signature value s without performing an operation of multiplying the signature value r by the secret key c. Therefore, the random integer k used to generate the signature value r and the secret key c used to generate the signature value s cannot be obtained through a side-channel attack.

Meanwhile, in an exemplary embodiment, the seed value generator710, the divider720, the first extractor730, the second extractor740, the third extractor750, and the calculator760, which are shown inFIG. 7, may be implemented on one or more computing devices including one or more processors and a computer-readable recording medium connected to the one or more processors. The computer-readable recording medium may be provided inside or outside the processor, and may be connected to the processor by various well-known means. The processor in the computing device may enable the computing device to operate according to the exemplary embodiments described in the specification. For example, the processor may execute instructions stored in the computer readable recording medium, and the instructions stored in the computer readable recording medium may cause a series of operational steps to be performed on the computing device according to the exemplary embodiments of the present disclosure described in the specification when executed by the processor.

FIG. 12is a flowchart illustrating a computation method according to an exemplary embodiment of the present disclosure.

The computation method shown inFIG. 12may be performed by the computation apparatus100illustrated inFIG. 1.

Referring toFIG. 12, the computation apparatus100generates a seed value (1210). According to an exemplary embodiment, the seed value may be formed by a random bit string.

Then, the computation apparatus100divides the generated seed value into a plurality of blocks (1220).

The computation apparatus100extracts a plurality of second parameter values that respectively correspond to the plurality of generated blocks from a second parameter candidate value set including a plurality of second parameter candidate values generated from a plurality of first parameter candidate values (1230).

In this case, according to an exemplary embodiment of the present disclosure, the second parameter candidate values included in the second parameter candidate value set may be values obtained by performing an exponentiation operation using each of the plurality of first parameter candidate values as an exponent or by performing a scalar multiplication operation of a point using each of the plurality of first parameter candidate values as a scalar multiplier.

In addition, according to an exemplary embodiment, the computation apparatus100may extract second parameter values respectively corresponding to bit strings in the plurality of generated blocks and positions of the blocks in the seed value, from the second parameter candidate value set.

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus100may generate the random number by multiplying or adding the second parameter values with each other.

Meanwhile, according to an exemplary embodiment of the preset disclosure, the computation apparatus100may encrypt data or generate a digital signature using the generated random number.

FIG. 13is a flowchart illustrating a computation method according to another exemplary embodiment of the present disclosure.

The computation method shown inFIG. 13may be performed by the computation apparatus400shown inFIG. 4.

Referring toFIG. 13, the computation apparatus400generates a seed value (1310). According to an exemplary embodiment, the seed value may be formed by a random bit string.

Then, the computation apparatus400divides the generated seed value into a plurality of blocks (1320).

The computation apparatus400extracts a plurality of first parameter values that respectively correspond to the plurality of generated blocks from a first parameter candidate value set including a plurality of first parameter candidate values (1330).

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus400may extract first parameter values respectively corresponding to bit strings in the plurality of generated blocks and positions of the blocks in the seed value from the first parameter candidate value set.

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus400may generate the first random number by adding the first parameter values to each other.

Then, the computation apparatus400extracts a plurality of second parameter values that respectively correspond to the plurality of generated blocks from a second parameter candidate value set including a plurality of second parameter candidate values generated using the plurality of first parameter candidate values included in the first parameter candidate value set (1350).

In this case, according to an exemplary embodiment of the present disclosure, the second parameter candidate values included in the second parameter candidate value set may be values obtained by performing an exponentiation operation using each of the plurality of first parameter candidate values included in the first parameter candidate value set as an exponent or by performing a scalar multiplication operation of a point using each of the plurality of first parameter candidate values as a scalar multiplier.

In addition, according to an exemplary embodiment, the computation apparatus400may extract second parameter values respectively corresponding bit strings in the plurality of generated blocks and positions of the blocks in the seed value, from the second parameter candidate value set.

According to an exemplary embodiment of the present disclosure, the computation apparatus400may generate the second random number by multiplying or adding the second parameter values with each other.

Meanwhile, according to an exemplary embodiment of the preset disclosure, the computation apparatus400may encrypt data or generate a digital signature using the generated first and second random numbers.

FIG. 14is a flowchart illustrating a computation method according to another exemplary embodiment of the present disclosure.

The computation method shown inFIG. 14may be performed by the computation apparatus700shown inFIG. 7.

Referring toFIG. 14, the computation apparatus700generates a seed value (1410). According to an exemplary embodiment, the seed value may be formed by a random bit string.

Then, the computation apparatus700divides the generated seed value into a plurality of blocks (1420).

Then, the computation apparatus700extracts a plurality of first parameter values that respectively correspond to the plurality of generated blocks from a first parameter candidate value set including a plurality of first parameter candidate values (1430).

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus700may extract first parameter values respectively corresponding to bit strings in the plurality of generated blocks and positions of the blocks in the seed value, from the first parameter candidate value set.

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus700may generate the first random number by adding the first parameter values to each other.

Then, the computation apparatus700extracts a plurality of second parameter values corresponding to the plurality of generated blocks from a second parameter candidate value set including the plurality of second parameter candidate values generated using each of the plurality of first parameter candidate values included in the first parameter candidate value set (1450).

In this case, according to an exemplary embodiment of the present disclosure, the second parameter candidate values included in the second parameter candidate value set may be values obtained by performing an exponentiation operation using each of the plurality of first parameter candidate values included in the first parameter candidate value set as an exponent or by performing a scalar multiplication operation of a point using each of the plurality of first parameter candidate values as a scalar multiplier.

In addition, according to an exemplary embodiment, the computation apparatus700may extract second parameter values respectively corresponding to bit strings in each of the plurality of generated blocks and positions of the blocks in the seed value, from the second parameter candidate value set.

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus700may generate the second random number by multiplying or adding the second parameter values with each other.

Thereafter, the computation apparatus700extracts a plurality of third parameter values corresponding to the plurality of generated blocks from a third parameter candidate value set including a plurality of third parameter candidate values generated using each of the plurality of second parameter candidate values included in the second parameter candidate value set (1470).

In this case, according to an exemplary embodiment of the present disclosure, at least some of the third parameter candidate values included in the third parameter candidate value set may include values obtained by multiplying at least some of the second parameter candidate values included in the second parameter candidate value set by a secret key.

In addition, according to an exemplary embodiment of the present disclosure, the computation apparatus700may extract third parameter values corresponding to bit strings in the plurality of generated blocks and positions of the blocks in the seed value, from the third parameter candidate value set.

In this case, according to an exemplary embodiment of the present disclosure, the computation apparatus700may generate the third random number by multiplying or adding the third parameter values with each other.

Meanwhile, according to an exemplary embodiment of the preset disclosure, the computation apparatus700may encrypt data or generate a digital signature using the generated first, second, and third random numbers.

While the flowcharts shown inFIGS. 12 to 14illustrate the method as being performed in a plurality of operations, at least some of the operations may be performed in a different order, performed in combination with each other, omitted, performed in sub-operations, or performed with at least one operation that is not illustrated being added thereto.

According to the exemplary embodiments of the present disclosure, it is possible to generate values equal to values obtained by performing 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.

Methods according to various exemplary embodiments of the present disclosure described above may be embodied as an application type that may be installed in electronic devices, i.e., IoT devices.

The methods according to the various exemplary embodiments of the present disclosure described above may also be embodied by merely upgrading software or hardware of electronic devices, i.e., IoT devices.

According to an exemplary embodiment, the elements, components, methods or operations described herein may be implemented using hardware components, software components, or a combination thereof. For example, the hardware components may include a processing device. According to an exemplary embodiment, the display apparatus may include a processing device, such as the image processor or the controller, that may be implemented using one or more general-purpose or special purpose computers, such as, for example, a hardware processor, a CPU, a hardware controller, an ALU, a DSP, a microcomputer, an FPGA, a PLU, a microprocessor or any other device capable of responding to and executing instructions in a defined manner. The processing device may run an operating system (OS) and one or more software applications that run on the OS. The processing device also may access, store, manipulate, process, and create data in response to execution of the software. For purpose of simplicity, the description of a processing device is used as singular; however, one skilled in the art will appreciated that a processing device may include multiple processing elements and multiple types of processing elements. For example, a processing device may include multiple processors or a processor and a controller. In addition, different processing configurations are possible, such a parallel processors.

Meanwhile, the exemplary embodiments of the present disclosure may include a computer readable recording medium including a program for executing methods described in this specification on a computer. The computer readable recording medium may include a program instruction, a local data file, a local data structure, and/or combinations and sub-combinations thereof. The medium may be specially designed and constructed for the purpose of the present disclosure, or may be well-known and available to those having skill in the computer software arts. Examples of the computer readable recording medium include magnetic media such as a hard disk, a floppy disk, and a magnetic tape, optical media such as a compact disc read-only memory (CD-ROM) and a digital versatile disc (DVD)-ROM, magneto-optical media such as a floptical disk, and a hardware device such as a read only memory (ROM), random-access memory (RAM), and a flash memory, which is specially designed to store and execute program commands. Examples of the program commands include an advanced language code which the computer can execute using an interpreter as well as a machine language code made by compilers.

Each of elements according to the above-described various exemplary embodiments (e.g., modules or programs) may include a single entity or a plurality of entities, and some of corresponding sub elements described above may be omitted or other types of sub elements may be further included in the various exemplary embodiments. Alternatively or additionally, some elements (e.g., modules or programs) may be integrated into one entity and then may equally or similarly perform a function performed by each of corresponding elements that are not integrated. Operations performed by modules, programs, or other types of elements according to the various exemplary embodiments may be sequentially, in parallel, or heuristically executed or at least some operations may be executed in different sequences or may be omitted, or other types of operations may be added.

While the present disclosure has been described in detail above with reference to representative exemplary embodiments, it should be understood by those skilled in the art that the exemplary embodiments may be variously modified without departing from the scope of the present disclosure. Therefore, the scope of the present disclosure is defined not by the described exemplary embodiments but by the appended claims and encompasses equivalents that fall within the scope of the appended claims.