Google Patentsopen access

Method for protecting data used in cloud computing with homomorphic encryption — Parallels IP Holdings GmbH (US8925075B2)

ABSTRACT

Abstract

A method for protection of cloud computing includes homomorphic encryption of data. Partially or fully homomorphic encryption allows for data within the cloud to be processed without decryption. A partially or fully homomorphic encryption is provided. The proposed scheme can be used with both an algebraic and analytical approaches. A cloud service is implemented on a server. A client encrypts data using fully homomorphic encryption and sends it to the server. The cloud server performs computations without decryption of the data and returns the encrypted calculation result to the client. The client decrypts the result, and the result coincides with the result of the same calculation performed on the initial plaintext data.

Description

CROSS-REFERENCE TO RELATED APPLICATION

This application is a non-provisional application of U.S. Provisional Patent Application No. 61/556,507, filed Nov. 7, 2011, which is incorporated by reference herein in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to data encryption methods, and more particularly, for protection of cloud system computations by homomorphic encryption.

2. Description of the Related Art

A conventional cloud is a computation resource that is provided to a user by a provider. All cloud services and infrastructure are implemented by the provider and are hidden from the user. The clouds can be divided into three classes:

Infrastructure as a Service—the provider implements functionality of Virtual Machines (VMs) and controls the infrastructure of the VMs that can be created, modified or deleted by the client (user). A user can decide how to use the Service. An example of such system is Amazon Web Services.

Platform as a Service—the provider provides to a user a certain platform for running applications. The provider is responsible for functionality of the platform and the user only provides an application to be launched on the platform. In this case, a cloud can be used for specialized tasks or types of tasks. An example of such system is Google App Engine platform.

Software as a Service—the provider provides a complete application for performing certain tasks. In this case, the provider is responsible for servicing the infrastructure, where the service runs, and for developing the application that supports the service. Such cloud has specific narrow specialization only for a particular task. An example of such system is Gmail.

Only the first two classes allow for running user application (i.e., logic) within the cloud. Protected cloud computing is a system where operations executed within the cloud cannot be determined from outside. Also, some data needs to be securely stored within the cloud for a long time. Additionally, transferring data into the cloud needs to be protected as well.

Thus, the protected cloud computing includes the following:

secure data storage within the cloud; secure data processing within the cloud; and secure data transfer in and out of the cloud.

Processing of data within the cloud is performed constantly. The cloud data often contains some critical personal data that can be accessed based on the technical limitation of the cloud. So, the only way to protect the data is to encrypt the files, so the perpetrator will not be able to read them.

Therefore, it is desired to have a system for protected cloud computing. Such system should have data in an encrypted form that cannot be read, even if the file is accessed. The data should be passed into the cloud already in the encrypted form. The encryption needs to be performed at the client site, so the cloud service does not know how the data is encrypted.

Also, the cloud should be able to process the encrypted data without decrypting it. Otherwise, the cloud becomes little more than a secure storage. Each operation on data would require sending the data back to the client for decryption and then sending the data back to the cloud for processing. Alternatively, an encryption key can be sent to the cloud for data decryption. However, this would jeopardize the data within the cloud.

Conventional cloud services do not provide completely secure data storage. In some rare cases the data can be encrypted on the client site. In other implementations the data is encrypted with the key that is stored in the same cloud. Both cases are not secure and are not convenient for efficient cloud computing. For example, personal or corporate taxes are calculated by third party services. It is obviously desired to not disclose some income information to the third parties. The personal data needs to be encrypted and sent over for processing. Then the results are received and decrypted.

Therefore, there is a need in the art to process encrypted data within the cloud without decryption. Accordingly, a method for homomorphic encryption of the cloud data is desired, so the data can be processed without decryption.

SUMMARY OF THE INVENTION

Accordingly, the present invention is related to a system and method for protection of cloud computing system by homomorphic encryption that substantially obviates one or more of the disadvantages of the related art.

A method for protection of cloud computing includes homomorphic encryption of data. Fully homomorphic encryption allows for data within the cloud to be processed without decryption.

Homomorphic encryption is a type of encryption where operations on unencrypted data have corresponding operations on encrypted data. If an operation O is performed on the encrypted data and then the data is decrypted, the result is the same as if the operation O has been performed on the unencrypted data.

According to the exemplary embodiment, a special scheme for homomorphic encryption is provided. The proposed scheme can be used with both algebraic and analytical approaches. A cloud service is implemented on a server. A client encrypts data and sends it to the server. The cloud server performs computations without decryption of the data and returns the encrypted calculation result to the client. The client decrypts the result and the result coincides with the result of the same calculation performed on the plaintext data.

Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned by practice of the invention. The advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE ATTACHED FIGURES

The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention.

In the drawings:

FIG. 1 illustrates a schematic of an exemplary computer system that can be used for implementation of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference will now be made in detail to the preferred embodiments of the present invention.

A method for homomorphic encryption of the cloud data is provided. Fully homomorphic encryption allows for data within the cloud to be processed without decryption. Fully homomorphic encryption schemes are malleable. A cryptosystem is malleable if it is possible to transform one ciphertext into another (in this case—math operations) without the decryption.

A principle of homomorphism can be described as follows:

Let ƒ: A→B where A and B are rings with addition, multiplication, and optionally zero and one.

Then, ƒ is homomorphism of the rings, if:

ƒ( a+ A b )=ƒ( a )+ B ƒ( b )

ƒ( a× A b )=ƒ( a )× B ƒ( b )

ƒ(1 A )−1 B

ƒ(0 A )=0 B

Homomorphism operations are secure (by definition), which is an inherent feature of any homomorphic encryption. Homomorphic encryption is a type of encryption where operations on unencrypted data have corresponding operations on encrypted data. If an operation O is performed on encrypted data and then the data is decrypted, the result is the same as if the operation O has been performed on unencrypted data.

Accordingly, fully homomorphic encryption provides the above condition for all operations. Otherwise, the encryption is partially homomorphic. Fully homomorphic encryption satisfies all of the requirements for encryption of protected cloud services. In other words, fully homomorphic encryption allows for encryption of data and performing operations on the encrypted data without decryption.

According to the exemplary embodiment, a special scheme for homomorphic encryption is provided. The proposed scheme can be used with both an algebraic and analytical approaches. The algebraic approach can be described as follows. Formal polynomials create a ring A[x] relative to addition and multiplication.

An example of homomorphism of the rings is as follows:

Transformation ƒ( x )= x+c,ƒ*:A[x]→A[x]   1.

A resulting (transformed) ƒ* generated from ƒ should be a homomorphism of the rings.

In case of formal polynomials (considered to have the same degree)

ƒ*( ā m + b

m )=ƒ*(( a 0 +b 0 ,a 1 +b 1 , . . . ,a m +b m ))

The corresponding resulting polynomial is

R ( x )=( a 0 +b 0 )+( a 1 +b 1 ) x + . . . +( a m +b m ) x m

while

ƒ( R ( x ))=( a 0 +b 0 )+( a 1 +b 1 )( x+c )+ . . . +( a m +b m )( x+c ) m

On the other hand, if P(x) and Q(x) correspond to formal polynomial ā (m) and b (m) , then

ƒ( P ( x ))+ƒ( Q ( x )= a 0 +a 1 ( x+c )+ . . . + a m ( x+c ) m +b 0 +b 1 ( x+c )+ . . . + b m ( x+c ) m

Accordingly, ƒ*(ā (k) + b (m) )=ƒ*(ā (k) )+ƒ*( b (m) )

A product can be described in a similar manner:

P ( x )· Q ( x )= a 0 b 0 +( a 0 b 1 +a 1 b 0 ) x ++( a 0 b 2 +a 1 b 1 +a 2 b 0 ) x 2 + . . . +a m b m x m+m

ƒ*( P ( x )· Q ( x ))= a 0 b 0 +( a 0 b 1 +a 1 b 0 )( x+c )++( a 0 b 2 +a 1 b 1 +a 2 b 0 )( x+c ) 2 + . . . +a m b m ( x+c ) m+m

Then,

ƒ*( P ( x ))·ƒ*( Q ( x ))= a 0 b 0 +( a 0 b 1 +a 1 b 0 )( x+c )++( a 0 b 2 +a 1 b 1 +a 2 b 0 )( x+c ) 2 + . . . +a m b m ( x+c ) m+m

Which coincides with ƒ*(P(x)·Q(x))

Accordingly, ƒ*(ā (k) · b (m) )=ƒ*(ā (k) )·ƒ*( b (m) )

Also, note that ƒ transforms 0 into 0, and 1 into 1.

Consequently, based on the definition, transformation, ƒ creates homomorphism of the rings ƒ* from ƒ. In other words

ƒ*( a+A[x]b )=ƒ*( a )+ A[x ]ƒ*( b )

Consider a transformation g(x)=c·x.

It is necessary to prove that g* transformed from g is also homomorphism of the rings.

A sum is:

⁢

g

*

⁡

(

R

⁡

(

x

)

)

=

a

0

+

b

0

+

(

a

1

+

b

1

)

⁢

cx

+

…

⁢

⁢

…

+

(

a

m

+

b

m

)

⁢

(

cx

)

m

.

⁢

g

*

⁡

(

P

⁡

(

x

)

CROSS-REFERENCE TO RELATED APPLICATION

This application is a non-provisional application of U.S. Provisional Patent Application No. 61/556,507, filed Nov. 7, 2011, which is incorporated by reference herein in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to data encryption methods, and more particularly, for protection of cloud system computations by homomorphic encryption.

2. Description of the Related Art

A conventional cloud is a computation resource that is provided to a user by a provider. All cloud services and infrastructure are implemented by the provider and are hidden from the user. The clouds can be divided into three classes:

Infrastructure as a Service—the provider implements functionality of Virtual Machines (VMs) and controls the infrastructure of the VMs that can be created, modified or deleted by the client (user). A user can decide how to use the Service. An example of such system is Amazon Web Services.

Platform as a Service—the provider provides to a user a certain platform for running applications. The provider is responsible for functionality of the platform and the user only provides an application to be launched on the platform. In this case, a cloud can be used for specialized tasks or types of tasks. An example of such system is Google App Engine platform.

Software as a Service—the provider provides a complete application for performing certain tasks. In this case, the provider is responsible for servicing the infrastructure, where the service runs, and for developing the application that supports the service. Such cloud has specific narrow specialization only for a particular task. An example of such system is Gmail.

Only the first two classes allow for running user application (i.e., logic) within the cloud. Protected cloud computing is a system where operations executed within the cloud cannot be determined from outside. Also, some data needs to be securely stored within the cloud for a long time. Additionally, transferring data into the cloud needs to be protected as well.

Thus, the protected cloud computing includes the following:

secure data storage within the cloud; secure data processing within the cloud; and secure data transfer in and out of the cloud.

Processing of data within the cloud is performed constantly. The cloud data often contains some critical personal data that can be accessed based on the technical limitation of the cloud. So, the only way to protect the data is to encrypt the files, so the perpetrator will not be able to read them.

Therefore, it is desired to have a system for protected cloud computing. Such system should have data in an encrypted form that cannot be read, even if the file is accessed. The data should be passed into the cloud already in the encrypted form. The encryption needs to be performed at the client site, so the cloud service does not know how the data is encrypted.

Also, the cloud should be able to process the encrypted data without decrypting it. Otherwise, the cloud becomes little more than a secure storage. Each operation on data would require sending the data back to the client for decryption and then sending the data back to the cloud for processing. Alternatively, an encryption key can be sent to the cloud for data decryption. However, this would jeopardize the data within the cloud.

Conventional cloud services do not provide completely secure data storage. In some rare cases the data can be encrypted on the client site. In other implementations the data is encrypted with the key that is stored in the same cloud. Both cases are not secure and are not convenient for efficient cloud computing. For example, personal or corporate taxes are calculated by third party services. It is obviously desired to not disclose some income information to the third parties. The personal data needs to be encrypted and sent over for processing. Then the results are received and decrypted.

Therefore, there is a need in the art to process encrypted data within the cloud without decryption. Accordingly, a method for homomorphic encryption of the cloud data is desired, so the data can be processed without decryption.

SUMMARY OF THE INVENTION

Accordingly, the present invention is related to a system and method for protection of cloud computing system by homomorphic encryption that substantially obviates one or more of the disadvantages of the related art.

A method for protection of cloud computing includes homomorphic encryption of data. Fully homomorphic encryption allows for data within the cloud to be processed without decryption.

Homomorphic encryption is a type of encryption where operations on unencrypted data have corresponding operations on encrypted data. If an operation O is performed on the encrypted data and then the data is decrypted, the result is the same as if the operation O has been performed on the unencrypted data.

According to the exemplary embodiment, a special scheme for homomorphic encryption is provided. The proposed scheme can be used with both algebraic and analytical approaches. A cloud service is implemented on a server. A client encrypts data and sends it to the server. The cloud server performs computations without decryption of the data and returns the encrypted calculation result to the client. The client decrypts the result and the result coincides with the result of the same calculation performed on the plaintext data.

Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned by practice of the invention. The advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed.

BRIEF DESCRIPTION OF THE ATTACHED FIGURES

The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and, together with the description, serve to explain the principles of the invention.

In the drawings:

FIG. 1 illustrates a schematic of an exemplary computer system that can be used for implementation of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference will now be made in detail to the preferred embodiments of the present invention.

A method for homomorphic encryption of the cloud data is provided. Fully homomorphic encryption allows for data within the cloud to be processed without decryption. Fully homomorphic encryption schemes are malleable. A cryptosystem is malleable if it is possible to transform one ciphertext into another (in this case—math operations) without the decryption.

A principle of homomorphism can be described as follows:

Let ƒ: A→B where A and B are rings with addition, multiplication, and optionally zero and one.

Then, ƒ is homomorphism of the rings, if:

ƒ( a+ A b )=ƒ( a )+ B ƒ( b )

ƒ( a× A b )=ƒ( a )× B ƒ( b )

ƒ(1 A )−1 B

ƒ(0 A )=0 B

Homomorphism operations are secure (by definition), which is an inherent feature of any homomorphic encryption. Homomorphic encryption is a type of encryption where operations on unencrypted data have corresponding operations on encrypted data. If an operation O is performed on encrypted data and then the data is decrypted, the result is the same as if the operation O has been performed on unencrypted data.

Accordingly, fully homomorphic encryption provides the above condition for all operations. Otherwise, the encryption is partially homomorphic. Fully homomorphic encryption satisfies all of the requirements for encryption of protected cloud services. In other words, fully homomorphic encryption allows for encryption of data and performing operations on the encrypted data without decryption.

According to the exemplary embodiment, a special scheme for homomorphic encryption is provided. The proposed scheme can be used with both an algebraic and analytical approaches. The algebraic approach can be described as follows. Formal polynomials create a ring A[x] relative to addition and multiplication.

An example of homomorphism of the rings is as follows:

Transformation ƒ( x )= x+c,ƒ*:A[x]→A[x]   1.

A resulting (transformed) ƒ* generated from ƒ should be a homomorphism of the rings.

In case of formal polynomials (considered to have the same degree)

ƒ*( ā m + b

m )=ƒ*(( a 0 +b 0 ,a 1 +b 1 , . . . ,a m +b m ))

The corresponding resulting polynomial is

R ( x )=( a 0 +b 0 )+( a 1 +b 1 ) x + . . . +( a m +b m ) x m

while

ƒ( R ( x ))=( a 0 +b 0 )+( a 1 +b 1 )( x+c )+ . . . +( a m +b m )( x+c ) m

On the other hand, if P(x) and Q(x) correspond to formal polynomial ā (m) and b (m) , then

ƒ( P ( x ))+ƒ( Q ( x )= a 0 +a 1 ( x+c )+ . . . + a m ( x+c ) m +b 0 +b 1 ( x+c )+ . . . + b m ( x+c ) m

Accordingly, ƒ*(ā (k) + b (m) )=ƒ*(ā (k) )+ƒ*( b (m) )

A product can be described in a similar manner:

P ( x )· Q ( x )= a 0 b 0 +( a 0 b 1 +a 1 b 0 ) x ++( a 0 b 2 +a 1 b 1 +a 2 b 0 ) x 2 + . . . +a m b m x m+m

ƒ*( P ( x )· Q ( x ))= a 0 b 0 +( a 0 b 1 +a 1 b 0 )( x+c )++( a 0 b 2 +a 1 b 1 +a 2 b 0 )( x+c ) 2 + . . . +a m b m ( x+c ) m+m

Then,

ƒ*( P ( x ))·ƒ*( Q ( x ))= a 0 b 0 +( a 0 b 1 +a 1 b 0 )( x+c )++( a 0 b 2 +a 1 b 1 +a 2 b 0 )( x+c ) 2 + . . . +a m b m ( x+c ) m+m

Which coincides with ƒ*(P(x)·Q(x))

Accordingly, ƒ*(ā (k) · b (m) )=ƒ*(ā (k) )·ƒ*( b (m) )

Also, note that ƒ transforms 0 into 0, and 1 into 1.

Consequently, based on the definition, transformation, ƒ creates homomorphism of the rings ƒ* from ƒ. In other words

ƒ*( a+A[x]b )=ƒ*( a )+ A[x ]ƒ*( b )

Consider a transformation g(x)=c·x.

It is necessary to prove that g* transformed from g is also homomorphism of the rings.

A sum is:

⁢

g

*

⁡

(

R

⁡

(

x

)

)

=

a

0

+

b

0

+

(

a

1

+

b

1

)

⁢

cx

+

…

⁢

⁢

…

+

(

a

m

+

b

m

)

⁢

(

cx

)

m

.

⁢

g

*

⁡

(

P

⁡

(

x

)

)

+

g

⁡

(

Q

⁡

(

x

)

)

=

a

0

+

a

1

⁡

(

cx

)

+

…

+

a

m

⁡

(

cx

)

m

++

⁢

b

0

+

b

1

⁡

(

cx

)

+

…

+

b

m

⁡

(

cx

)

m

Accordingly, g*(ā (k) + b (m) )=g*(ā (k) )+g*( b (m) )

The product is:

g

*

⁡

(

P

⁡

(

x

)

·

Q

⁡

(

x

)

)

=

a

0

⁢

b

0

+

(

a

0

⁢

b

1

+

a

1

⁢

b

0

)

⁢

(

cx

)

++

⁢

(

a

0

⁢

b

2

+

a

1

⁢

b

1

+

a

2

⁢

b

0

)

⁢

(

cx

)

2

+

…

+

a

m

⁢

b

m

⁡

(

cx

)

m

+

m

⁢

Then

g

*

⁡

(

P

⁡

(

x

)

)

·

g

⁡

(

Q

⁡

(

x

)

)

=

a

0

⁢

b

0

+

(

a

0

⁢

b

1

+

a

1

⁢

b

0

)

⁢

(

cx

)

++

⁢

(

a

0

⁢

b

2

+

a

1

⁢

b

1

+

a

2

⁢

b

0

)

⁢

(

cx

)

2

+

…

+

a

m

⁢

b

m

⁡

(

cx

)

m

+

m

Which corresponds to g*(P(x)·Q(x))

Accordingly, g*(ā (k) b (m) )=g*(ā (k) )g*( b (m) )

Thus, transformation of g transforms 0 into 0, and 1 into 1.

Therefore, transformation g creates homomorphism of the rings g*

g *( a+A[x]b )= g *( a )+ A[x]g *( b ).

Any polynomial P(x)=a 0 +a 1 x+a 2 x 2 + . . . +a m x m creates a homomorphism P* rings of a formal polynomial. Thus:

P*: A[x]→A[x]

P *( a+A[x]b )= P *( a )+ A[x]P *( b )  1.

P *( a·A[x]b )= P *( a )· A[x]P *( b )  2.

P *(0 A[x] )=0 A[x]   3.

P *(1 A[x] )=1 A[x]   4.

Homomorphic encryption in case Z 2 /[x].

Let the ring have only two elements 0 and 1.

Two elements need to be encrypted based on the above algorithm for performing operations on them. An element z 1 has a formal polynomial (z 1 , a 1 , a 2 , . . . , a m ). Application of homomorphism P results in a formal polynomial (q 0 , q 1 , q 2 , . . . , q m+p ).

Encryption of z 2 results in (r 0 , r 1 , r 2 , . . . , r m+p ).

Accordingly, after the operations are performed, a formal polynomial is produced.

This polynomial, once decrypted, gives the result in a first position. If operations are performed on polynomials (both polynomials are the results of encryption of z 1 and z 2 ) in the form z 1 +ƒ 1 (z 1 )+ƒ 2 (z 1 ) and z 2 +g 1 (z 2 )+g 2 (z 2 ), then a multiplication and addition will result in something like (z 1 *z 2 +z 1 )+(ƒ 1 (z 1 )*g 1 (z 2 )+ƒ 1 (z 1 ))+ . . . .

The first element in the brackets can be used as the result of the operation, if the “tail” is deleted from data that is not significant, from the perspective of the task that is being performed. This example shows that functionality, in the proposed method, is achieved by making the computations more complex and by increasing the amount of computation necessary, even for simple arithmetic operations. Multiplication and addition can be viewed as operations on matrices, and the size of the matrix becomes apparent during decryption, which is needed to identify the meaningful portion of the element.

For example, in case of the operation q m+p + q m+p · r m+p , after decryption, the first position in the resulting formal polynomial will have z 1 +z 1 ·z 2 .

For example, to find value m 1 +m 2 bits of polynomials (m 1 ,a,b), (m 2 ,c,d) are compared. Homomorphism created by the polynomial P(x)=p+q·x is applied. The resulting polynomials are:

( m 1 +ap+bp 2 ,aq+ 2 bpq,bq 2 )

( m 2 +cp+dp 2 ,cq+ 2 dpq,dq 2 )

Adding them results in:

R =( m 1 +m 2 +ap+cp+bp 2 +dp 2 ,aq+cq+ 2 bpq+ 2 dpq,bq 2 +dq 2 )

In order to decrypt, the polynomial

R ⁡

( x )

=

m 1

+

m 2

+ ap + cp +

bp 2

+

dp 2

++

⁢

(

aq + cq +

2 ⁢

⁢ bpq

+

2 ⁢

⁢ dpq

)

⁢ x

+

(

bq 2

+

dq 2

)

⁢

x 2

is divided by p+qx, which gives the remainder m 1 +m 2 , as a result.

In case of multiplication, the brackets are not open and the formal polynomial looks like:

m 1 +a ( p+qx )+ b ( p+xq ) 2

m 2 +c ( p+qx )+ d ( p+xq ) 2

Multiplication of the encrypted polynomials results in:

m

1

⁢

m

2

+

(

m

1

⁢

c

+

m

2

⁢

a

)

⁢

(

p

+

qx

)

++

⁢

(

m

1

⁢

d

+

ac

+

m

2

⁢

b

)

⁢

(

p

+

qx

)

2

++

⁢

(

ad

+

bc

)

⁢

(

p

+

qx

)

3

+

bd

⁡

(

p

+

qx

)

4

Note that if the brackets are open, it will not be possible to recreate anything without knowing p+qx.

However, if the resulting polynomial divided by p+qx the needed bit is produced.

An analytical approach can be described as follows. Consider a class of functions over set M with values located on the ring A.

These functions create a ring F=A(M) relative to discrete addition and multiplication operations. Let G=A(S), where S is a mathematical set. Consider transformation φ: M→S. This transformation can be called a variable substitution—x=φ(y), xεS, yεM. Such a substitution of variables creates homomorphism φ* of the rings of functions: φ*: F→G.

This can be proven as follows. Let ƒ(x), g(x) be functions of ring F and φ*(ƒ(x)), φ*(g(x)) be functions of ring G.

f

⁡

(

x

)

+

g

⁡

(

x

)

→

φ

*

⁡

(

f

⁡

(

x

)

+

g

⁡

(

x

)

)

==

f

⁡

(

φ

⁡

(

x

)

)

+

g

⁡

(

φ

⁡

(

x

)

)

==

φ

*

⁡

(

f

⁡

(

x

)

)

+

φ

*

⁡

(

g

⁡

(

x

)

)

.

1

f

⁡

(

x

)

·

g

⁡

(

x

)

→

φ

*

⁡

(

f

⁡

(

x

)

·

g

⁡

(

x

)

)

==

f

⁡

(

φ

⁡

(

x

)

)

·

g

⁡

(

φ

⁡

(

x

)

)

==

φ

*

⁡

(

f

⁡

(

x

)

)

·

φ

*

⁡

(

g

⁡

(

x

)

)

.

2

⁢

1

F

→

1

G

.

3

⁢

0

F

→

0

G

.

4

Thus, φ* is really a homomorphism of the rings of the functions.

One example can use polynomials of rings of real numbers, integers, or prime numbers.

R[x]—a ring of polynomials.

Any polynomial P(x) creates homomorphism P*: R[x]→R[x].

In a client-server model, this can be implemented as follows. A client wants to perform calculations on the server in such a way that the server cannot know what data is involved in the calculations. For example, a client wants to calculate a value of a polynomial function ƒ(x 1 , x 2 , . . . x n ) at a point (a 1 , a 2 , . . . a n ). The following steps are executed:

a secret key x 0 is randomly picked on the client site, x 0 is a real number;

for each number a i from a vector of values selected b i , c i , such that b i x 0 +c i =a i ;

linear polynomials are sent to the server in a form of b i x+c i along with a polynomial function ƒ(x 1 , . . . x n );

the client requests the server to substitute x i with linear polynomials. The server substitutes the polynomials into the function ƒ:

ƒ( b 1 x+c 1 , . . . ,b n x+c n ).

Then, the server opens the brackets and sends the result to the client. Thus, the client receives the coefficients of the polynomial that assumes the desired value at a point x 0 .

The client substitutes x 0 into the resulting polynomial function, and finds the desired value. Note that the server does not know the data value, for which the client wants to calculate the values of the function.

This scheme of homomorphic encryption allows to protect operations and to sort the encrypted data. Since the operations are performed with real numbers, the point x 0 has an area where lines corresponding to a i x+b i do not intersect. Then a correct order is kept at point x 0 , but it remains unchanged in the area.

The use of real numbers permits taking advantage of orthogonality of the transformations, in other words, different initial data will always give different encryption results. Also, for any pair of encrypted texts strings, it is possible to identify a point x 0 , such that in its neighborhood, linear functions derived from decryption results maintain their relative values (smaller-larger) comparison, both before and after encryption and decryption. For each pair, there is a particular point x 0 , that can be identified without decryption. The use of these points x 0 permits sorting of the list, where the list does not have identical elements.

Accordingly, an element can be given to the server without disclosing the encryption key.

The following schemes for fully homomorphic encryption are used in the exemplary embodiment.

Domingo-Ferre Scheme.

Two prime numbers are selected—p and q, n=pq. A positive integer d is also selected. Then (d,n) is an open key. Then, from Z p and Z q , elements r p and r q are used, that create large multipliable sub-groups in Z p and Z q , respectively. Then (p, q, r p , r q ) is a secret key.

<div num="p-0094

CLAIMS

Claims ( 25 )

What is claimed is:

1. A system for protected cloud computing, the system comprising:

a server receiving data from a client, wherein the data is provided to the server in encrypted form; and

a cloud service implemented on the server for performing calculations for the client, wherein the server performs calculations without decryption of the data and provides a result back to the client, such that the client can decrypt the result,

wherein:

a finite set of initial elements are generated on the client and transformed into a set of encrypted elements by applying a partially or fully homomorphic encryption algorithm; and

resulting encrypted elements belong to the finite set of the initial elements and each of the encrypted elements has only one corresponding initial element,

wherein the initial elements are transformed into the encrypted elements by a first encryption algorithm using a secret key z 0 represented by n bits, for any initial element u of the Galois field GF(2 n )[x], there are n randomly generated elements a 1 , a 2 , . . . , a n of the Galois field for which

a 0 =u −( a 1 z 0 +a 2 ( z 0 ) 2 + . . . +a d ( z 0 ) n ),

p is a prime number, a 0 , a 1 , . . . , a n are coefficients of a polynomial v=a 0 +a 1 x+a 2 x 2 + . . . +a d x n that corresponds to u, and

element v is encrypted into element u by u=a 0 +a 1 z 0 +a 2 (z 0 ) 2 + . . . +a d (z 0 ) n

wherein a set of coefficients of a polynomial used as an encryption key, and

wherein the encryption uses an encryption polynomial h(x)=ξ+(x−λ)*r(x), where r(x) is an arbitrary polynomial from a Galois field G(2 n )[x], and ξ and λ are fixed elements of the Galois field G(2 n )[x].

2. The system of claim 1 , wherein Galois field elements are used as the initial elements.

3. The system of claim 2 , wherein the Galois field elements are prime numbers.

4. The system of claim 2 , wherein the Galois field elements are vectors.

5. The system of claim 2 , wherein the Galois field elements correspond to real numbers.

6. The system of claim 1 , wherein a result of an operation on any pair of the encrypted elements produces a resulting encrypted element belonging to the set of the encrypted elements.

7. The system of claim 1 , wherein the encryption is partially homomorphic.

8. The system of claim 1 , wherein the encryption is fully homomorphic.

9. The system of claim 1 , wherein a result of the calculations performed on the server is encrypted and can only be used after decryption with a secret key.

10. The system of claim 1 , wherein each operation on the encrypted elements has only one corresponding operation on the initial elements, and the operation produces a resulting initial element corresponding to the resulting encrypted element.

11. The system of claim 1 , wherein a result of the calculations performed on the server is homomorphically encrypted and can only be used after decryption with a secret key.

12. The system of the claim 1 , wherein the initial elements are polynomials with coefficients from a Galois field.

13. The system of claim 12 , wherein each initial element is a bit set, wherein each bit is a coefficient of the Galois field polynomial.

14. The system of claim 12 , wherein the encryption comprises a direct transformation of elements of a first Galois field into elements of a second Galois field.

15. The system of claim 12 , wherein the encryption comprises a direct transformation of elements of a first Galois field into elements of a different representation of the first Galois field.

16. The system of claim 1 , wherein the encrypted elements are polynomials, and all operations are performed by the cloud service on the encrypted elements, including any multiplication operations, without increasing a degree of the polynomials.

17. The system of claim 1 , wherein the encrypted elements are polynomials, and all operations are performed by the cloud service on the encrypted elements, including any multiplication operations, result in polynomials having a degree no higher than original polynomials.

18. The system of claim 1 , wherein the encrypted elements are polynomials, and all operations are performed by the cloud service on the encrypted elements, including any multiplication operations, result in polynomials having a degree that is a constant known prior to the operations.

19. A system for protected cloud computing, the system comprising:

a server receiving data from a client, wherein the data is provided to the server in encrypted form; and

a cloud service implemented on the server for performing calculations for the client, wherein the server performs calculations without decryption of the data and provides a result back to the client, such that the client can decrypt the result,

wherein:

a finite set of initial elements are generated on the client and transformed into a set of encrypted elements by applying a partially or fully homomorphic encryption algorithm; and

resulting encrypted elements belong to the finite set of the initial elements and each of the encrypted elements has only one corresponding initial element,

wherein a set of coefficients of a polynomial used as an encryption key, and

wherein the encryption uses an encryption polynomial h(x)=ξ+(x−λ)*r(x), where r(x) is an arbitrary polynomial from a Galois field G(2 n )[x], and ξ and λ are fixed elements of the Galois field G(2 n )[x], and

wherein a reverse transformation uses a decryption algorithm and a secret key z 0 represented by n bits, for any initial element u of the Galois field GF(2 n )[x], there are n randomly generated elements a 1 , a 2 , . . . , a n of the Galois field for which

a 0 =u −( a 1 z 0 +a 2 ( z 0 ) 2 + . . . +a d ( z 0 ) n ),

p is a prime number, a 0 , a 1 , . . . , a n are coefficients of a polynomial v=a 0 +a 1 x+a 2 x 2 + . . . +a d x n that corresponds to u, and

element v is encrypted into element u by u=a 0 +a 1 z 0 +a 2 (z 0 ) 2 + . . . +a d (z 0 ) n .

20. The system of claim 19 , wherein a reverse transformation uses a decryption algorithm and a secret key z 0 represented by n bits, for any initial element u of the Galois field GF(2 n )[x], there are n randomly generated elements a 1 , a 2 , . . . , a n of the Galois field for which

a 0 =u −( a 1 z 0 +a 2 ( z 0 ) 2 + . . . +a d ( z 0 ) n ),

p is a prime number, a 0 , a 1 , . . . , a n are coefficients of a polynomial v=a 0 +a 1 x+a 2 x 2 + . . . +a d x n that corresponds to u, and

element v is encrypted into element u by u=a 0 +a 1 z 0 +a 2 (z 0 ) 2 + . . . +a d (z 0 ) n .

21. A computer-implemented method for protected cloud computing, the method comprising configuring at least one computing device to perform the functions of:

encrypting data on a client device using a secret key in a form of coefficients of a polynomial uses an encryption polynomial h(x)=ξ+(x−λ)*r(x), where r(x) is an arbitrary polynomial from a Galois field G(2 n )[x] and ξ and λ are fixed elements of the Galois field G(2 n )[x];

sending the coefficients and the encrypted data for calculations to a cloud service implemented on a server; and

receiving a result of the cloud service calculations and decrypting the result on the client and using the secret key,

wherein the data is encrypted by a fully homomorphic encryption using the coefficients of the polynomial as the secret key applied to the data, the data being in a form of polynomials,

wherein the fully homomorphic encryption comprises a direct transformation of polynomials of a first Galois field into polynomials of a second Galois field, the first and second Galois fields being Galois rings, and

wherein the data is transformed into the encrypted elements by a first encryption algorithm using a secret key z 0 represented by n bits, for any initial element u of the Galois field GF(2 n )[x], there are n randomly generated elements a 1 , a 2 , . . . , a n of the Galois field for which

a 0 =u −( a 1 z 0 +a 2 ( z 0 ) 2 + . . . +a d ( z 0 ) n ),

p is a prime number, a 0 , a 1 , . . . , a n are coefficients of a polynomial v=a 0 +a 1 x+a 2 x 2 + . . . +a d x n that corresponds to u, and

element v is encrypted into element u by u=a 0 + 1 z 0 +a 2 (z 0 ) 2 + . . . +a d (z 0 ) n .

22. A computer-implemented method for processing homomorphically encrypted data, the method comprising configuring at least one computing device to perform the functions of:

in a Galois field, defining a set of initial elements;

defining at least one operation in the Galois field such that a result of the operation also belongs to the Galois field, wherein the operation is any of addition and multiplication;

using a secret key, defining parameters for transforming the initial elements into a second set of coefficients representing a power polynomial, such that the second set of coefficients also belong to the Galois field, and it possible to recover a corresponding element of the Galois field by only using the secret key; and

starting a session on a user side for processing the data,

wherein

selected initial elements are transformed into the coefficients by using the parameters;

for each selected initial element, transmitting the coefficients to a server, together with an order of one or more operations and a description of using the result of the operation;

performing the operation on the server side on the second set of elements that also belong to the Galois field;

transmitting the results of the operation on the server to the client; and

using the secret key to recover the result from the Galois field on the client side;

wherein

the secret key is element z 0 belonging to the Galois field GF(p n ), with z 0 represented by a length n bits, for any initial element u of the Galois field GF(p n ), there are n randomly generated elements a 1 , a 2 , . . . , a n of the Galois field for which

a 0 =u −( a 1 z 0 +a 2 ( z 0 ) 2 + . . . +a d ( z 0 ) n ),

p is a prime number, a 0 , a 1 , . . . , a n are coefficients of a polynomial v=a 0 +a 1 x+a 2 x 2 + . . . +a d x n that corresponds to u,

element v is encrypted into element u by u=a 0 +a 1 z 0 +a 2 (z 0 ) 2 + . . . +a d (z 0 ) n , and

wherein the polynomials are operated on using polynomial arithmetic, and

coefficients of the resulting polynomials are homomorphically calculated from the coefficients of the polynomials with the Galois field operations, based on rules of the operations in the Galois field.

23. The method of claim 22 , wherein the coefficients are generated from a random number k, with 2&lt;k&lt;n, and a secret set of elements s ij εGF(p n ) i=1, 2, . . . , k, j=k+1, . . . , n, which are the same for the session, for both client and server, and wherein

the coefficients a 1 , a 2 , . . . , a k are selected based on a i =s i1 a k+1 +s i2 a k+2 + . . . +s id a d , i=1, 2, . . . , k, and

the coefficients a k+1 , a k+2 , . . . , a n are randomly selected.

24. The method of claim 22 , wherein p=2.

25. The method of claim 22 , wherein p elements λ 1 , λ 2 , . . . λ p of the Galois Field and the polynomial r(x) of a degree k are used to generate a polynomial s(x) of a degree k+p, s(x) in a form of s(x)=(x−λ 1 )(x−λ 2 ) . . . (x−λ p )r(x);

element λ 0 is selected the set λ 1 , λ 2 , . . . λ p and is used as the secret key;

u being the data on which server operations are performed using a polynomial f(x)=a 0 +a 1 x+a 2 x 2 + . . . +a k+p−1 x k+p−1 of a degree k+p−1, such that

u=f (λ 0 )= a 0 +a 1 λ 0 +a 2 (λ 0 ) 2 + . . . +a k+p−1 (λ 0 ) k+p−1 ,

wherein u and s(x) are transmitted to the server,

all calculations are performed in the ring GF(2 n )[x]/s(x), and

a degree of any resulting polynomial is no higher than k+p−1.

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