KR20180013064A - Homomorphic Encryption Method of a Plurality of Messages Supporting Approximate Arithmetic of Complex Numbers - Google Patents

Homomorphic Encryption Method of a Plurality of Messages Supporting Approximate Arithmetic of Complex Numbers Download PDF

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KR20180013064A
KR20180013064A KR1020160096184A KR20160096184A KR20180013064A KR 20180013064 A KR20180013064 A KR 20180013064A KR 1020160096184 A KR1020160096184 A KR 1020160096184A KR 20160096184 A KR20160096184 A KR 20160096184A KR 20180013064 A KR20180013064 A KR 20180013064A
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polynomial
encryption method
present
homomorphic encryption
message
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KR101861089B1 (en
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천정희
송용수
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서울대학교산학협력단
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/008—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols involving homomorphic encryption
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F21/00—Security arrangements for protecting computers, components thereof, programs or data against unauthorised activity
    • G06F21/60—Protecting data

Abstract

A homomorphic encryption method according to the present invention is performed by a computer, and comprises: a first step of using a canonical embedding function to convert a plurality of message vectors (m) into polynomial expression (f) for satisfying mathematic formula, f(ζ_i) = m_i; and a second step of performing homomorphic encryption on the polynomial expression converted in the first step. In mathematic formula, ζ_i is a different solution of n^th integer irreducible polynomial, Φ(x); Φ(x) ∈ Ζ[x], in a complex number. The purpose of the present invention is to provide a homomorphic encryption method which uses an approximate complex number operation to enable an arrangement of a plurality of messages.

Description

근사 복소수 연산을 지원하는 복수 개의 메시지의 동형 암호화 방법{Homomorphic Encryption Method of a Plurality of Messages Supporting Approximate Arithmetic of Complex Numbers}[0001] The present invention relates to a method of encrypting a plurality of messages supporting approximate complex numbers,

본 발명은 동형 암호화 방법에 관한 것으로서 좀 더 자세하게는 복수 개의 메시지를 동형 암호화할 뿐 아니라 유효 숫자 연산까지 지원할 수 있는 동형 암호화 방법에 관한 것이다.BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an encryption method of the same type, and more particularly, to an encryption method capable of not only encrypting a plurality of messages of the same type but also supporting an effective number operation.

최근에는 서버에 개인 정보 내지 자료를 저장하고 필요한 경우에 사용자가 사용자 단말기를 통해서 서버에 저장되어 있는 데이터를 이용하도록 하는 클라우드 컴퓨팅 환경이 날로 늘어나고 있다. 개인 정보 내지 데이터를 서버에 보관할 때에는 데이터 유출을 방지하는 등 개인 정보 보호를 위해 데이터를 암호화하여 보관하게 된다. 이처럼 암호화된 데이터가 서버에 보관되어 있는 경우 해당 데이터를 검색하거나 연산을 통한 소정의 작업을 할 때에 암호화된 데이터를 일일이 복호화한 후에 원하는 검색 또는 연산을 수행하여야 하기 때문에 매우 비효율적이고, 연산을 위해 일시적으로 복호화된 개인 정보 내지 자료가 제3자에게 유출될 가능성이 증대되는 단점이 존재한다.Recently, a cloud computing environment in which personal information or data is stored in a server and data is stored in a server through a user terminal when necessary is increasing. When storing personal information or data on a server, data is encrypted and stored for personal information protection, such as preventing data leakage. When the encrypted data is stored in the server, it is very inefficient because the encrypted data must be decrypted and then subjected to a desired search or calculation when searching the corresponding data or performing a predetermined operation through the operation. There is a disadvantage that the possibility that the personal information or data decrypted by the third party is leaked to the third party is increased.

이러한 문제를 해결하기 위해 동형 암호화 방법이 널리 연구되고 있다. 동형 암호화에 의하면, 암호화된 정보나 자료를 복호화하지 않고 암호문 자체에 대해서 연산을 해도 평문에 대해 동일한 연산을 수행한 후 암호화한 결과와 동일한 결과를 얻기 때문에 암호문을 복호화하지 않고도 소정의 연산을 수행할 수 있다.To solve this problem, homogeneous encryption methods are widely studied. According to the homotypic encryption, since the same operation is performed on the plain text even if the cipher text itself is computed without decrypting the encrypted information or data, the same result as the result of the encryption is obtained, so that a predetermined operation is performed without decrypting the cipher text .

동형 암호화에 있어서 유효숫자 연산을 지원하면서 복수 개의 메시지를 동시에 처리하는 배치(batch) 처리는 가능하지 않은 영역이었다.In the same type of encryption, batch processing that simultaneously processes a plurality of messages while supporting valid numeric operations was not possible.

본 발명은 전술한 종래 기술의 문제를 해결하기 위하여 근사 복소수 연산을 이용하여 복수 개의 메시지의 배치 처리가 가능한 동형 암호화 방법을 제공하는 것을 목적으로 한다.SUMMARY OF THE INVENTION It is an object of the present invention to provide an isochronous encryption method capable of batch processing a plurality of messages using approximate complex numbers to solve the problems of the prior art described above.

본 발명에 의한 동형 암호화 방법은 컴퓨터에 의해서 수행되며, 복수 개의 메시지 벡터(m)를 기본 매장 함수(canonical embedding function)를 이용하여 다음 수학식을 만족하는 다항식 f로 변환하는 제1 단계와; 제1 단계에서 변환된 다항식을 동형 암호화하는 제2 단계를 포함한다.The method of the present invention includes a first step of performing a method of encrypting a plurality of message vectors m into a polynomial f satisfying the following equation using a canonical embedding function; And a second step of encrypting the transformed polynomial in the first step.

<수학식>&Lt; Equation &

Figure pat00001
Figure pat00001

여기에서

Figure pat00002
는 복소수 내에서 n차 정수 기약 다항식(
Figure pat00003
;
Figure pat00004
)의 서로 다른 해이다.From here
Figure pat00002
Is an n-th order integer irreducible polynomial in the complex number (
Figure pat00003
;
Figure pat00004
).

제2 단계는 Ring-LWE 기반 동형 암호화 방법에 의해서 동형 암호화하는 단계가 될 수 있다.The second step can be a step of encrypting the same by the Ring-LWE-based isochronous encryption method.

제2 단계는, 다항식 f를 실수 동형 암호화 방법에 의해서 동형 암호화하는 단계가 될 수 있다.The second step may be a step of encrypting the polynomial f by the real method homogeneous encryption method.

상기 n차 정수 기약 다항식은 사이클로토믹 다항식(cyclotomic polynomial)인 것이 바람직하다.The n-th order integer irreducible polynomial is preferably a cyclotomic polynomial.

본 발명에 의하면, 복수 개의 복소수 메시지쌍을 다항식으로 변환하는 과정을 통해서 하나의 암호문으로 암호화할 수 있는 작용 효과가 제공된다. 그리고 암호화된 다항식간의 덧셈과 곱셉이 메시지 벡터간의 연산에 대응하므로 동형 암호화의 특성도 그대로 유지할 수 있다.According to the present invention, an effect of encrypting a single ciphertext through a process of converting a plurality of complex message pairs into a polynomial is provided. Since the addition and multiplication between encrypted polynomials correspond to the operation between message vectors, the characteristics of homogeneous encryption can be maintained.

나아가 전술한 바와 같이 유효숫자 연산도 지원하는 동형 암호화 방법으로서 기능하는 작용 효과도 있다.Furthermore, as described above, there is also an operational effect that functions as an isochronous encryption method that also supports an effective numerical operation.

도 1은 본 발명에 의한 동형 암호화 방법의 순서도.1 is a flowchart of an isochronous encryption method according to the present invention;

이하에서는 본 발명에 대해서 자세하게 설명한다.Hereinafter, the present invention will be described in detail.

본 발명에 의한 암호화 방법 및 연산은 전자적 연산 기능을 가지는 컴퓨터와 같은 전자 장치에 의해서 수치해석적 방법에 따라 수행되며, 그러한 전자 장치의 종류는 비제한적인 의미로 정의된다. 공지되어 있는 수치해석적 방법 중에서 본 발명이 제안하는 수식에 적용 가능한 것을 사용하면 되며, 구체적인 수치해석적 방법은 본 발명의 본질과 관계가 없기 때문에 자세한 설명은 생략한다.The encryption method and calculation according to the present invention are performed according to a numerical analysis method by an electronic device such as a computer having an electronic computing function, and the kind of such electronic device is defined in a non-limiting sense. Of the known numerical analytical methods, those applicable to the formulas proposed by the present invention may be used, and detailed numerical analytical methods are not related to the essence of the present invention, and therefore, detailed description thereof will be omitted.

그리고 본 발명에 의한 동형 암호화 알고리즘은 기존에 공지되어 있는 동형 암호화 방법 중 본 발명의 목적을 달성하기에 적합한 것이라면 어느 것을 이용해도 무방하며, 동형 암호화 알고리즘 각각의 구체적인 설명은 본 발명의 본질과 거리가 있기 때문에 설명의 편의성 및 간략화를 위해서 본 명세서에 구체적으로 설명하지는 않는다.The same type of encryption algorithm according to the present invention may be used as long as it is suitable for achieving the object of the present invention among the known encryption methods known in the art, Therefore, the present invention is not specifically described herein for convenience and simplicity of explanation.

본 발명의 발명자는 복수 개(n개)의 메시지를 동시에(배치 처리) 동형 암호화하는 방법을 가능하게 하기 위한 방법으로서 먼저 복소수 (

Figure pat00005
) 내에서 서로 다른 근
Figure pat00006
을 가지는 n차 정수 기약다항식(irreducible polynomial;
Figure pat00007
Figure pat00008
Figure pat00009
)을 선택한다. 동형 암호화에서 복소수 개념을 도입하는 것은 본 발명자가 최초이며 이러한 복소수 개념을 도입함으로써 후술하는 바와 같이 복수 개의 메시지를 동시에 동형 암호화하는 것이 가능하다.The inventors of the present invention have proposed a method for simultaneously encrypting a plurality of (n) messages (batch processing)
Figure pat00005
),
Figure pat00006
(N) is an irreducible polynomial.
Figure pat00007
Figure pat00008
Figure pat00009
). The present inventor is the first to introduce the concept of complex numbers in the same type of encryption, and by introducing such a complex number concept, it is possible to simultaneously encrypt a plurality of messages as described later.

다음으로 기본 매장함수(canonical embedding function;

Figure pat00010
)를 계산한다. 기본 매장함수는 다항식(
Figure pat00011
)를
Figure pat00012
의 근
Figure pat00013
에서의 값들의 쌍
Figure pat00014
로 대응시키는 함수이다. 이 함수가 동형함수(homomorphism)임은 이 분야에 평균적 지식을 가진 자라면 쉽게 증명할 수 있다.Next, the canonical embedding function
Figure pat00010
). The default store function is a polynomial (
Figure pat00011
)
Figure pat00012
Near
Figure pat00013
A pair of values at
Figure pat00014
. This function is homomorphism, and it is easy to prove it to anyone with average knowledge in this field.

기본 매장함수를 행렬(C)로 표현하면 다음과 같다.The basic store function is represented by matrix (C) as follows.

Figure pat00015
Figure pat00015

다항식 f(x)를 계수들의 열벡터

Figure pat00016
The polynomial f (x) is the column vector of coefficients
Figure pat00016

로 표현하면 이 다항식의 canonical embedding인

Figure pat00017
와는 다음과 같은 관계 즉
Figure pat00018
를 만족하게 된다.Canonical embedding of this polynomial
Figure pat00017
And the following relation
Figure pat00018
.

Figure pat00019
Figure pat00019

다음으로 도 1을 참조하여 본 발명에 의한 동형 암호화 방법에 대해서 설명한다.Next, the same type encryption method according to the present invention will be described with reference to FIG.

먼저 단계(100)에서 복수 개(n개)의 메시지 벡터(m = (m1,...,mn))를 입력받는다. 그리고 기본 매장 함수를 이용하여 입력받은 메시지 벡터를 다음과 같은 수학식에 의해 다항식(f(x))으로 변환한다.(120)First, at step 100, a plurality of (n) message vectors m = (m 1 , ..., m n ) are received. Then, the input message vector is transformed into a polynomial f (x) by the following equation using the basic store function (120)

Figure pat00020
Figure pat00020

변환된 다항식 f(x)는 다음과 같은 관계를 만족한다.The transformed polynomial f (x) satisfies the following relation.

Figure pat00021
Figure pat00021

메시지 벡터를 변환한 다항식에 대해서 다항식간의 연산을 지원하는 동형암호 방법으로 다항식 f(x)를 동형암호한다.(140) 그러한 동형 암호 방법 중 하나인 Ring LWE 문제에 기반하는 스킴을 이용하여 동형 암호화하면 암호문은 (a, as + f + e)의 형태로 산출된다. a는 랜덤한 다항식, s는 비밀키, e는 크기가 작은 에러 다항식이다. 본 명세서에서는 Ring LWE 문제에 기반하는 스킴을 이용하여 동형 암호화한 실시예에 대해서 설명하지만 Ring LWE 문제에 기반하는 스킴 이외에도 다항식간의 연산을 지원하는 동형 암호 방법이라면 어느 것을 사용해도 무방하다.The polynomial f (x) is homologous to the polynomial cryptosystem, which supports operations between polynomials for the polynomial transforming the message vector. (140) Using the scheme based on the Ring LWE problem, one of such homogeneous cryptosystems, The ciphertext is computed in the form (a, as + f + e). a is a random polynomial, s is a secret key, and e is a small error polynomial. In this specification, the same encryption scheme using the scheme based on the Ring LWE problem is described, but any scheme that supports the operation between polynomials in addition to the scheme based on the Ring LWE problem can be used.

상기 암호문을 복호화하면 다항식(f + e)을 얻을 수 있다. 이 다항식을 전술한 기본 매장 함수를 이용하여 메시지 벡터 형태로 변환하면 다음과 같은 순서쌍을 얻을 수 있다.When decrypting the cipher text, a polynomial (f + e) can be obtained. By transforming this polynomial into the message vector form using the above basic store function, the following ordered pair can be obtained.

Figure pat00022
Figure pat00022

n차 정수 기약다항식(

Figure pat00023
)의 근
Figure pat00024
의 크기가 작고 다항식 e의 계수들이 작다면 벡터
Figure pat00025
의 크기 역시 충분히 작기 때문에 복호화하면 메시지 벡터(m)를 얻을 수 있다.nth order irreducible polynomial (
Figure pat00023
)
Figure pat00024
Is small and the coefficients of the polynomial e are small,
Figure pat00025
The size of the message vector m is sufficiently small so that the message vector m can be obtained by decoding.

위 요구 사항을 만족하는 n차 정수 기약다항식(

Figure pat00026
)으로 cyclotomic polynomial을 선택하는 것이 바람직하다. n차 정수 기약다항식을 cyclotomic polynomial로 선택하면 기약다항식의 해(
Figure pat00027
)는 다음과 같은 조건을 만족한다.The nth order integer irreducible polynomial satisfying the above requirement (
Figure pat00026
), It is preferable to select a cyclotomic polynomial. If the nth-order integer irreducible polynomial is chosen as the cyclotomic polynomial, then the solution of the irreducible polynomial (
Figure pat00027
) Satisfy the following conditions.

Figure pat00028
Figure pat00028

이렇게 되면 벡터

Figure pat00029
의 크기는
Figure pat00030
보다 작기 때문에 전술한 조건을 만족한다.Then,
Figure pat00029
The size of
Figure pat00030
The above-mentioned condition is satisfied.

한편, 스킴의 효율성을 위해서는 메시지 벡터를 변환한 다항식 f의 크기가 작은 것이 바람직하다.On the other hand, for the efficiency of the scheme, it is desirable that the size of the polynomial f transforming the message vector is small.

전술한 다항식 f를 산출하는 수학식에 의하면 메시지 벡터의 크기가 작고, C-1의 크기가 작다면 다항식 f의 크기 역시 작다. 기약다항식이 cyclotomic polynomial인 경우에는 C-1의 크기가 작다는 것은 알려져 있다.According to the equation for calculating the polynomial f described above, if the size of the message vector is small and the size of C -1 is small, the size of the polynomial f is also small. It is known that the magnitude of C -1 is small when the irreducible polynomial is a cyclotomic polynomial.

전술한 바와 같이 기본 매장 함수(

Figure pat00031
)가 동형 성질을 가지기 때문에 다음과 같이 다항식간의 연산은 메시지 벡터간의 연산에 대응되므로, 본 발명에 의한 동형 암호화 방법은 동형 암호화의 특성에 부합한다.As described above, the basic store function (
Figure pat00031
) Has the same property, the operation between polynomials corresponds to the operation between message vectors as follows, so that the isochronous encryption method according to the present invention conforms to the characteristics of the same type of encryption.

Figure pat00032
Figure pat00032

Figure pat00033
(
Figure pat00034
는 벡터의 성분 각각을 곱하는 연산을 의미함)
Figure pat00033
(
Figure pat00034
Means an operation of multiplying each of the components of the vector)

본 발명의 발명자는 이 출원에 앞서 유효 숫자 연산을 지원하는 동형 암호문의 생성 방법에 대한 특허출원(특허출원 제10-2016-0075859호)을 하고 인터넷(http://eprint.iacr.org/2016/421)에 논문으로 발표한 바 있다. 이러한 동형 암호화문의 유효숫자 연산을 하기 위해서는 메시지 크기(mangitude significand)를 유지하기 위하여 메시지를 반올림(rounding)하는 과정이 필요한데, modulus-switching 방법을 통해서 수행하는 것이 바람직하다. 이 방법을 간단하게 설명하면 (mod q)상의 다항식을 더 작은 (mod q')상의 다항식으로 변환하는 방법인데 주어진 다항식에 q'/q를 곱하고 가까운 정수 다항식으로 반올림하는 과정을 거친다.Prior to this application, the inventor of the present invention has filed a patent application (Patent Application No. 10-2016-0075859) on a method of generating an isochronous ciphertext to support an effective numerical operation, and the Internet (http://eprint.iacr.org/2016 / 421). In order to perform the effective numeric operation of the same type of encryption algorithm, it is necessary to round the message in order to maintain the message size (mangitude significance). It is desirable to perform the rounding through the modulus-switching method. This method is briefly described as a method of converting a polynomial on (mod q) into a polynomial on a smaller (mod q ') polynomial by multiplying q' / q by a given integer polynomial.

전술한 다항식

Figure pat00035
은 메시지 벡터(m)에 대응하는 다항식이고 다항식 f를 modulus-switching 방법을 통해서 작은 다항식 (f/p)로 변형시키면, 이에 대응하는 메시지는
Figure pat00036
이 된다. 반올림(rounding) 과정에서 추가적으로 생기는 에러 다항식은 메시지에 비해서 충분히 작기 때문에 무시할 수 있다. 결국, 본 발명에 의한 동형 암호화 방법에 의하더라도 전술한 특허출원의 유 효 숫자 연산이 가능한 작용효과가 제공된다.The polynomial
Figure pat00035
Is a polynomial corresponding to the message vector m and transforming the polynomial f into a small polynomial f / p through a modulus-switching method, the corresponding message is
Figure pat00036
. The additional error polynomial in the rounding process is negligible since it is small enough for the message. As a result, even if the isochronous encryption method according to the present invention is applied, the effect of enabling efficient numerical calculation of the above-mentioned patent application is provided.

본 발명에 의하면, 복수 개의 복소수 메시지쌍을 다항식으로 변환하는 과정을 통해서 하나의 암호문으로 암호화할 수 있는 작용 효과가 제공된다. 그리고 암호화된 다항식간의 덧셈과 곱셉이 메시지 벡터간의 연산에 대응하므로 동형 암호화의 특성도 그대로 유지할 수 있다. 복소수 메시지쌍의 예로는, 이미지를 실수 행렬에 대응시킨 후에 Singular Value Decomposition을 하면 eigenvector와 eigenvalue로 분해되는데, 이 때 복소수 메시지가 될 수 있으며 이러한 복소수 메시지쌍을 본 발명에 의한 방법으로 암호화할 수 있다. 이러한 이미지 암호화뿐만 아니라 Cyber Physical System, 사물 인터넷의 센싱 정보, 개인 건강 관리(Private Health Care)의 개인건강정보를 암호화하는데에도 사용할 수 있다.According to the present invention, an effect of encrypting a single ciphertext through a process of converting a plurality of complex message pairs into a polynomial is provided. Since the addition and multiplication between encrypted polynomials correspond to the operation between message vectors, the characteristics of homogeneous encryption can be maintained. An example of a complex number message pair is that when an image is mapped to a real number matrix and then a Singular Value Decomposition is performed, it is decomposed into an eigenvector and an eigenvalue, which can be a complex number message and the complex number message pair can be encrypted by the method of the present invention . In addition to encrypting these images, it can also be used to encrypt personal health information of Cyber Physical System, sensing information of things Internet, and private health care.

나아가 전술한 바와 같이 유효숫자 연산도 지원하는 동형 암호화 방법으로서 기능하는 작용 효과도 있다.Furthermore, as described above, there is also an operational effect that functions as an isochronous encryption method that also supports an effective numerical operation.

이상 첨부 도면을 참고하여 본 발명에 대해서 설명하였지만 본 발명의 권리범위는 후술하는 특허청구범위에 의해 결정되며 전술한 실시예 및/또는 도면에 제한되는 것으로 해석되어서는 아니된다. 그리고 특허청구범위에 기재된 발명의, 당업자에게 자명한 개량, 변경 및 수정도 본 발명의 권리범위에 포함된다는 점이 명백하게 이해되어야 한다.While the present invention has been described with reference to the accompanying drawings, it is to be understood that the scope of the present invention is defined by the claims that follow, and should not be construed as limited to the above-described embodiments and / or drawings. It is to be expressly understood that improvements, changes and modifications that are obvious to those skilled in the art are also within the scope of the present invention as set forth in the claims.

Claims (4)

컴퓨터에 의해서 수행되는 동형 암호화 방법에 있어서,
복수 개의 메시지 벡터(m)에 대해서 다음 수학식을 만족하는 다항식 f로 변환하는 제1 단계와,
제1 단계에서 변환된 다항식을 동형 암호화하는 제2 단계를 포함하는,
동형 암호화 방법.
<수학식>
Figure pat00037

여기에서
Figure pat00038
는 복소수 내에서 n차 정수 기약 다항식(
Figure pat00039
;
Figure pat00040
)의 서로 다른 해임.
In a homogeneous encryption method performed by a computer,
A first step of transforming a plurality of message vectors m into a polynomial f satisfying the following equation;
And a second step of homologously encrypting the transformed polynomial in the first step,
Homogeneous encryption method.
&Lt; Equation &
Figure pat00037

From here
Figure pat00038
Is an n-th order integer irreducible polynomial in the complex number (
Figure pat00039
;
Figure pat00040
).
청구항 1에 있어서,
제2 단계는 Ring-LWE 기반 동형 암호화 방법에 의해서 동형 암호화하는,
동형 암호화 방법.
The method according to claim 1,
The second step is to encrypt the same by the Ring-LWE based homogeneous encryption method,
Homogeneous encryption method.
청구항 1에 있어서,
제2 단계는, 다항식 f를 실수 동형 암호화 방법에 의해서 동형 암호화하는,
동형 암호화 방법.
The method according to claim 1,
In the second step, the polynomial f is isomorphically encrypted by the real-
Homogeneous encryption method.
청구항 1 내지 청구항 3 중 어느 하나의 청구항에 있어서,
n차 정수 기약 다항식은 사이클로토믹 다항식(cyclotomic polynomial)인,
동형 암호화 방법.


The method according to any one of claims 1 to 3,
The nth order integer irreducible polynomial is a cyclotomic polynomial,
Homogeneous encryption method.


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