CN113420314A - Fully homomorphic encryption algorithm suitable for high-speed operation - Google Patents

Abstract

The invention discloses a fully homomorphic encryption algorithm suitable for high-speed operation, which comprises the following steps: firstly, randomly selecting two lambda-bit large prime numbers p and q for a safety parameter input by a system, wherein the public parameter is N ═ pq; and a second step of encryption: for plaintext data m_i，i∈[1,n]Encrypting to obtain cipher text C suitable for addition calculation according to specific formula_i,sumAnd ciphertext C suitable for addition calculation_i,mulAnd calculates a random number r_iR of_sumAnd r_mul(ii) a Thirdly, carrying out ciphertext operation, namely, carrying out ciphertext C operation according to a specific formula_i,sumPerforming addition operation or on ciphertext C_i,mulPerforming multiplication operation; fourthly, decrypting the ciphertext subjected to the addition operation according to a specific formula to obtain M_sumOr decrypting the ciphertext subjected to multiplication to obtain M_mul. The invention can reduce the storage space occupied by the ciphertext obtained by encrypting the plaintext data, realize high-efficiency encryption and high-efficiency operation, and ensure that the safety can reach the unconditional safety level.

Description

Fully homomorphic encryption algorithm suitable for high-speed operation

Technical Field

The invention relates to the technical field of fully homomorphic encryption algorithms, in particular to a fully homomorphic encryption algorithm suitable for high-speed operation.

Background

A fully homomorphic encryption algorithm refers to an encryption method that has a fully homomorphic nature. The ciphertext generated by the algorithm supports mathematical operation, and the result of the operation based on the ciphertext is the same as the result of the operation based on the ciphertext after decryption.

The classical fully homomorphic encryption algorithm has three main problems:

1. the efficiency is low: traditional homomorphic encryption is based on a lattice cipher design, has a large number of matrix operations, and is bit-by-bit encrypted. This results in a high complexity of the execution time of the algorithm, which is difficult to be applied in practical engineering.

2. And (3) ciphertext expansion: the ciphertext produced by the homomorphic encryption is a plurality of large matrices, while the plaintext input is bits. Therefore, fully homomorphic encryption can expand the storage space occupied by data by thousands of times. The storage efficiency is seriously lowered.

3. The safety problem is as follows: fully homomorphic encryption because the ciphertext is malleable, the security of fully homomorphic encryption can only reach CPA level, but not CCA security and CCA2 security.

How to improve the operation speed in the fully homomorphic encryption algorithm, simultaneously prevent the expansion of the ciphertext and ensure the safety of the ciphertext is the problem to be solved at present.

Disclosure of Invention

The invention aims to provide a fully homomorphic encryption algorithm suitable for high-speed operation, a ciphertext obtained by the algorithm can be directly used for mathematical operation, and meanwhile, the occupied storage space is small.

In order to achieve the purpose, the invention provides the following technical scheme: an all homomorphic encryption algorithm suitable for high speed operations, comprising the steps of:

firstly, randomly selecting two lambda-bit large prime numbers p and q for a safety parameter input by a system, wherein the public parameter is N ═ pq;

and a second step of encryption:

when a data owner wishes to have a batch of plaintext data m_i，i∈[1,n]When encrypting and adding the encrypted ciphertext by using the cloud service, the data m is encrypted_iEncryption is performed as follows:

C_i，sum＝m_i+r_imod N, to obtain ciphertext C suitable for addition computation_i,sum，

Obtained all for addingCiphertext C of arithmetic operation_i,sumAfter that, the following calculation is performed:

when the data owner wishes to outsource the batch of data onto the cloud for multiplication, data m is multiplied_iEncryption is performed as follows:

C_i,mul＝m_i·r₁mod N, to obtain ciphertext C suitable for addition computation_i,mul，

Obtaining ciphertext C all for multiplication_i,mulAfter that, the following calculation is performed:

wherein r in the encryption operation process_iIs a random number, the resulting ciphertext C_i,sumAnd ciphertext C_i,mulCan be directly used for mathematical operation;

thirdly, carrying out ciphertext following operation:

after receiving the encrypted data sent by the data provider, the cloud service performs operation in the following manner:

and (3) addition:

multiplication:

fourthly, decrypting the result of the ciphertext operation:

after the server finishes the operation, the operation ciphertext is returned to the data provider, the data provider decrypts the operation ciphertext, and after the decryption is successful, the data provider can obtain a calculation result, wherein the decryption process is as follows:

for the addition decryption result: m_sum＝C_sum-r_sum mod N；

For the multiplication result, r is first pair as follows_mulInversion:

then, the decryption operation is performed as follows, so that the multiplication decryption calculation result can be obtained:

preferably, in the second step, the random number r needs to be selected again when each plaintext in the data batch is encrypted_i。

Compared with the prior art, the invention has the following beneficial effects:

the invention provides an efficient fully homomorphic mapping which can be combined with other cryptography technologies to form a fully homomorphic encryption scheme, and can realize efficient encryption and efficient operation; in consideration of ciphertext expansion, ciphertext-free expansion can be realized theoretically; security can also reach an unconditional security level.

Detailed Description

The technical solutions of the present invention will be described clearly and completely with reference to the following embodiments of the present invention, and it should be understood that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.

The invention provides a technical scheme that: an all homomorphic encryption algorithm suitable for high speed operations, comprising the steps of:

firstly, randomly selecting two lambda-bit large prime numbers p and q for a safety parameter input by a system, wherein the public parameter is N ═ pq;

and a second step of encryption: for plaintext data m_i，i∈[1,n]Encrypting to obtain cipher text C suitable for addition calculation according to specific formula_i,sumAnd ciphertext C suitable for addition calculation_i,mulAnd calculates a random number r_iR of_sumAnd r_mul；

Thirdly, carrying out ciphertext operation, namely, carrying out ciphertext C operation according to a specific formula_i,sumPerforming addition operation or on ciphertext C_i,mulPerforming multiplication operation;

fourthly, decrypting the ciphertext subjected to the addition operation according to a specific formula to obtain M_sumOr decrypting the ciphertext subjected to multiplication to obtain M_mul。

In the second step of this embodiment, when the data owner wishes to have a batch of plaintext data m_i，i∈[1,n]When encrypting and adding the encrypted ciphertext by using the cloud service, the data m is encrypted_iEncryption is performed as follows:

C_i，sum＝m_i+r_imod N, to obtain ciphertext C suitable for addition computation_i,sum，

Obtaining ciphertext C all for addition_i,sumAfter that, the following calculation is performed:

when the data owner wishes to outsource the batch of data onto the cloud for multiplication, data m is multiplied_iEncryption is performed as follows:

C_i,mul＝m_i·r₁mod N, to obtain ciphertext C suitable for addition computation_i,mul，

Obtaining ciphertext C all for multiplication_i,mulAfter that, the following calculation is performed:

wherein r in the encryption operation process_iIs a random number, ciphertext C_i,sumAnd ciphertext C_i,mulCan be directly used for mathematical operation;

in the third step of this embodiment:

after receiving the encrypted data sent by the data provider, the cloud service performs ciphertext operation as follows:

and (3) addition:

multiplication:

in the fourth step of this embodiment:

after the server finishes the operation, the operation ciphertext is returned to the data provider, the data provider decrypts the operation ciphertext, and after the decryption is successful, the data provider can obtain a calculation result, wherein the decryption process is as follows:

for the addition decryption result: m_sum＝C_sum-r_sum mod N；

For the multiplication result, r is first pair as follows_mulInversion:

then, the decryption operation is performed as follows, so that the multiplication decryption calculation result can be obtained:

in the present embodiment, in the second step, each plaintext m in the batch of data_iThe random number r needs to be reselected when encryption is carried out_iFurther improving the plaintext m_iSecurity of the ciphertext obtained after encryption.

Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.

Claims (2)

1. A fully homomorphic cryptographic algorithm suitable for high speed operations, comprising the steps of:

firstly, randomly selecting two lambda-bit large prime numbers p and q for a safety parameter input by a system, wherein the public parameter is N ═ pq;

and a second step of encryption:

when a data owner wishes to have a batch of plaintext data m_i，i∈[1,n]When encrypting and adding the encrypted ciphertext by using the cloud service, the data m is encrypted_iEncryption is performed as follows:

C_i，sum＝m_i+r_imod N, to obtain ciphertext C suitable for addition computation_i,sum，

Obtaining ciphertext C all for addition_i,sumAfter that, the following calculation is performed:

when the data owner wishes to outsource the batch of data onto the cloud for multiplication, data m is multiplied_iEncryption is performed as follows:

C_i,mul＝m_i·r₁mod N, to obtain ciphertext C suitable for addition computation_i,mul，

Obtaining ciphertext C all for multiplication_i,mulAfter that, the following calculation is performed:

wherein the encryption operation processR in_iIs a random number;

thirdly, carrying out ciphertext following operation:

after receiving the encrypted data sent by the data provider, the cloud service performs operation in the following manner:

and (3) addition:

multiplication:

fourthly, decrypting the result of the ciphertext operation:

after the server finishes the operation, the operation ciphertext is returned to the data provider, the data provider decrypts the operation ciphertext, and after the decryption is successful, the data provider can obtain a calculation result, wherein the decryption process is as follows:

for the addition decryption result: m_sum＝C_sum-r_summod N；

For the multiplication result, r is first pair as follows_mulInversion:

then, the decryption operation is performed as follows, so that the multiplication decryption calculation result can be obtained:

2. the fully homomorphic encryption algorithm for high speed operations according to claim 1, wherein: in the second step, each plaintext in the data block needs to be encrypted by re-selecting the random number r_i。