CN103475472B - The full homomorphic cryptography method of NTRU types on ring LWE - Google Patents
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Abstract
The invention discloses a kind of full homomorphic cryptography methods of NTRU types on ring LWE, by defined parameters；Carry out the ciphering process of a part homomorphism；Proceed without the full homomorphic cryptography process of startup.It is exchanged and mould switching technology by key, with addition encryption and multiplication encryption in ciphering process, the noise for exporting ciphertext is less than the noise of input ciphertext, play the role of updating ciphertext, but its efficiency realized will be due to the process of startup, to realize the full homomorphic cryptography method for being not necessarily to start, encryption efficiency is high, and the ciphertext of acquisition is also shorter.
Description
Technical field
The present invention relates to encryption method field, the full homomorphic cryptography method of NTRU types on specifically a kind of ring LWE.
Background technology
Full homomorphic cryptography can carry out ciphertext the calculating of arbitrary function, this is special in the case where not knowing key
Property so that full homomorphic cryptography is widely used demand, such as；Cloud computing safety, data base ciphertext search, Secure meter
Calculation, ciphertext data programmable system etc..Full homomorphic cryptography is applied under cloud computing environment, user can be by encrypted data
It is contracted out to high in the clouds, then high in the clouds can be according to the request (such as inquiry, statistics etc.) of user, and it (is to ciphertext to do corresponding calculate
Carry out, the encryption of general type cannot calculate ciphertext, and only full homomorphic cryptography just can be with), then result is returned
Back to user, desired result can be obtained by after user's decryption.User is because the data of oneself are encrypted without being exposed to
High in the clouds, data-privacy safety are ensured that high in the clouds can do ciphertext arbitrary calculation process because being not necessarily to decryption, from
And realize the safety of cloud computing.Full homomorphic cryptography research is had important section by the very big concern of academia and industrial quarters
Learn meaning and application value.
Full homomorphic cryptography was just proposed by Rivest, Adleman and Dertouzos early in 1978, just became password later
The open problem of one of educational circles, is known as " Holy grail " of password educational circles.Then many homomorphic encryption schemes were generated in succession, these
Or scheme is to meet additive homomorphism or is to meet multiplicative homomorphic, more also it can meet limited sub-addition simultaneously and multiply
The homomorphism scheme of method, but neither one is that (" complete " of full homomorphism refers to that arbitrary function can be counted full homomorphism
It calculates).Until Gentry in 2009 it is breakthrough construct first full homomorphic encryption scheme.Gentry is in circuit counting model
Under, full homomorphic encryption scheme is constructed based on ideal lattice.Although full homomorphic encryption scheme realizes, the scheme of Gentry has
Two defects：First is that (progressive complexity is about inefficient), wherein λ is security parameter)；Second is the mistake of structural scheme
Two relied in journey assume (circulation safe problem and sparse subset and problem) not by people's in-depth study mistake.So
The scheme of Gentry is like one piece with white jade defective, although having breakthrough meaning, there is also some problems.
The building method of Gentry is divided into three steps：The first step constructs a Somewhat homomorphic encryption scheme, i.e., can only hold
Row limited number of time multiplication and additional calculation；Second step, compression decryption circuit, that is, simplify decryption circuit so that decrypt the depth of circuit
The circuit depth that can be executed less than Somewhat homomorphisms scheme；Third walks, startup scheme, i.e., after each cryptogram computation, to close
Literary homomorphism executes decryption circuit, as the noise of gained ciphertext remain in a fixed size, be equivalent to noise reduction.
The full homomorphic encryption scheme of the first generation follows the initial building methods of Gentry.The full homomorphic encryption scheme of the second generation is based on
LWE problem or RLWE problems, structural form are simpler.Especially in BGV schemes, an effective Noise management is introduced
Technology：Mould switching technology so that without starting the noise that can about subtract ciphertext.After its principle is each ciphertext product, to close
Literary vector is multiplied by a scale factor and is reduced.Carrying for the noise of the exponential multiplication number of plies can be obtained using mould switching technology
Height is combined with Internet Key Exchange, can obtain the full homomorphic encryption scheme of hierarchical without starting.
NTRU encipherment schemes are earliest one of encipherment schemes, and with efficiently famous, therefore are constructed complete same on NTRU
State encipherment scheme is very significant, and the prior art is not based on the full homomorphic encryption scheme on NTRU.Based on above for complete same
The analysis of state encryption method, the full homomorphic cryptography method of the prior art also there is also encryption efficiencies poor, the too long of defect of ciphertext.
Invention content
The technical problem to be solved by the present invention is to provide NTRU types on the ring LWE that a kind of encryption efficiency is high, ciphertext is shorter
Full homomorphic cryptography method.
Technical solution of the invention is to provide the full homomorphic cryptography method of NTRU types on the ring LWE of following steps, packet
Include following steps：
One, defined parameters：
For integer q, definitionEncryption beAnd Rq=R/qR is enterprising
Capable, wherein φ (x)=χn+ 1 is cyclotomic polynomial, and n is 2 power side, q be prime number and
If polynomial f ∈ R and satisfaction | | f | |∞≤ B, then it is the boundaries B to claim f；
{ χn(n ∈ N) be distributed collection on a R, if for arbitrary f ← χnHave | | f | |∞≤ B, then claim { χnIt is B
Boundary is distributed, i.e. the boundaries B distribution one boundaries B multinomial of output on a R；
Gaussian ProfileIt has the following properties that：For, arbitrary real numberHave：
Two, the ciphering process of a part homomorphism is carried out；
(1) parameter is established：Select μ mould q and n=(λ, μ) and Gaussian Profile χ=χ (λ, μ) so that these parameters
It can ensure and obtain 2 on ring LWEλSafety；It enablesParameter params=(q, n, χ)；
(2) generation of key：F ' ← χ is chosen, calculates f ← 2f '+1 so that f ≡ 1 (mod2)；If f is in RqIn be irreversible
, then f ', setting private key sk=f ∈ R are chosen again；
(3) generation of public key：Choose g ← χ, setting public key pk=h=2gf-1∈Rq；
(4) it encrypts：Choose s, e ← χ, output ciphertext c ← hs+2e+m ∈ RqEven if it is close that public key adds an information m to obtain
Literary c；
(5) it decrypts：Calculate μ ← fc ∈ Rq；Export m ← μm od2；
Due to fc=fhs+2fe+fm=2gs+2fe+fm, if | | fc | |∞＜ q/2, then μ=fc, μ (mod2)=fm
(mod2)=m, as long as so meeting | | fc | |∞＜ q/2, then above-mentioned steps are correct；
Three, the full homomorphic cryptography process of startup is proceeded without；
(1) parameter is established：L is the number of plies of circuit, enables μ=μ (λ, L)=θ (log λ+logL), and to j=0 ..., L is obtained
The mode sequence q to successively decrease0..., qL, the identical Gaussian Profile χ of each layer of use and ringParameter
paramsj={ qj, λ, χ, n } and (j=0 ..., L)；
(2) generation of public key：Key f is generated in the method for the second step of step 2j, walked with the third of step 2
Method generate public key hj；It enablesTo f 'jAnd fj+1Key exchange is carried out, is enabledKey sk includes fj,
Public key pk includes hjWithWherein j=0 ..., L does not have ζ, each layer of circuit to have corresponding public key and private as j=L
Key triple
(3) it encrypts：An information m is added to be encrypted public key pk；
(4) it decrypts；Assuming that ciphertext c is in jth layer circuit, corresponding key fj, private key sk is decrypted to obtain in plain text
M,
(5) additive process：Assuming that c1、c2Decruption key be fj, i.e., in same layer circuit, if not same layer circuit can
To carry out key exchange；It enablesc3Key be fj, by c3Key be set asThat is f 'jAgain by about subtracting
The method of ciphertext noise obtains new ciphertext c4；
(6) multiplication process：Assuming that c1、c2Decruption key be fj, i.e., in same layer circuit, if not same layer circuit can
To carry out key exchange；It enablesc3Key beAgain by about subtracting the method for ciphertext noise,
Obtain new ciphertext c4。
Method using the present invention, compared with prior art, the present invention has the following advantages：The present invention is exchanged by key
With mould switching technology, with addition encryption and multiplication encryption in ciphering process, the noise for exporting ciphertext is less than input ciphertext
Noise plays the role of updating ciphertext, but its efficiency realized will be not necessarily to startup due to the process of startup to realize
Full homomorphic cryptography method, encryption efficiency is high, and the ciphertext of acquisition is also shorter.
As an improvement, the method for about subtracting ciphertext noise includes two steps：The first step is that key exchanges, and ciphertext is converted
At the ciphertext of next layer of circuit, key byIt is transformed into fj+1, c1Corresponding key is fj+1, corresponding mould is qj, the second stepping
Row mould exchanges, and about subtracts the noise of ciphertext, c2Corresponding key is fj+1, corresponding mould is qj+1；fj+1Value is smaller, is selected from height
This distribution χ, to ensure that the validity of mould exchange.
As an improvement, it includes two processes that the key, which exchanges,：First process is two keys of input and mould, output
Auxiliary information is to ensure to exchange；Second process is input auxiliary information and initial ciphertext, exports a new ciphertext, initial ciphertext
It is to same encryption of plaintext with new ciphertext.
Specific implementation mode
With regard to specific embodiment, the invention will be further described below.
The full homomorphic cryptography method of NTRU types, includes the following steps on the ring LWE of the present invention：
One, defined parameters：
For integer q, definitionEncryption beAnd Rq=R/qR is enterprising
Capable, wherein φ (x)=χn+ 1 is cyclotomic polynomial, and n is 2 power side, q be prime number and
If polynomial f ∈ R and satisfaction | | f | |∞≤ B, then it is the boundaries B to claim f；
{ χn(n ∈ N) be distributed collection on a R, if for arbitrary f ← χnHave | | f | |∞≤ B, then claim { χnIt is B
Boundary is distributed, i.e. the boundaries B distribution one boundaries B multinomial of output on a R；
Gaussian ProfileIt has the following properties that：ForArbitrary real numberHave：
Two, the ciphering process of a part homomorphism is carried out；
(1) parameter is established：Select μ mould q and n=(λ, μ) and Gaussian Profile χ=χ (λ, μ) so that these parameters
It can ensure and obtain 2 on ring LWEλSafety；It enablesParameter params=(q, n, χ)；
(2) generation of key：F ' ← χ is chosen, calculates f ← 2f '+1 so that f ≡ 1 (mod2)；If f is in RqIn be irreversible
, then f ', setting private key sk=f ∈ R are chosen again；
(3) generation of public key：Choose g ← χ, setting public key pk=h=2gf-1∈Rq；
(4) it encrypts：Choose s, e ← χ, output ciphertext c ← hs+2e+m ∈ RqEven if it is close that public key adds an information m to obtain
Literary c；
(5) it decrypts：Calculate μ ← fc∈Rq；Export m ← μm od2；
Due to fc=fhs+2fe+fm=2gs+2fe+fm, if | | fc | |∞＜ q/2, then μ=fc, μ (mod2)=fm
(mod2)=m, as long as so meeting | | fc | |∞＜ q/2, then above-mentioned steps are correct；
Three, the full homomorphic cryptography process of startup is proceeded without；
(1) parameter is established：L is the number of plies of circuit, enables μ=μ (λ, L)=θ (log λ+logL), and to j=0 ..., L is obtained
The mode sequence q to successively decrease0..., qL, the identical Gaussian Profile χ of each layer of use and ringParameter
paramsj={ qj, λ, χ, n } and (j=0 ..., L)；
(2) generation of public key：Key f is generated in the method for the second step of step 2j, walked with the third of step 2
Method generate public key hj；It enablesIt is rightAnd fj+1Key exchange is carried out, is enabledKey sk includes fj, public
Key pk includes hjWithWherein j=0 ..., L does not have ζ, each layer of circuit to have corresponding public key and private key as j=L
Triple
(3) it encrypts：An information m is added to be encrypted public key pk；
(4) it decrypts；Assuming that ciphertext c is in jth layer circuit, corresponding key fj, private key sk is decrypted to obtain in plain text
M,
(5) additive process：Assuming that c1、c2Decruption key be fj, i.e., in same layer circuit, if not same layer circuit can
To carry out key exchange；It enablesc3Key be fj, by c3Key be set asI.e.Again by about subtracting
The method of ciphertext noise obtains new ciphertext c4；
(6) multiplication process：Assuming that c1、c2Decruption key be fj, i.e., in same layer circuit, if not same layer circuit can
To carry out key exchange；It enablesc3Key beAgain by about subtracting the method for ciphertext noise, obtain
Go out new ciphertext c4。
The method for about subtracting ciphertext noise includes two steps：The first step is that key exchanges, and ciphertext is converted into next layer
The ciphertext of circuit, key byIt is transformed into fj+1, f1Corresponding key is fj+1, corresponding mould is qj, second step progress mould exchange,
About subtract the noise of ciphertext, f2Corresponding key is fj+1, corresponding mould is qj+1；fj+1Value is smaller, is selected from Gaussian Profile χ, from
And it ensure that the validity that mould exchanges.
It includes two processes that the key, which exchanges,：First process is two keys of input and mould, exports auxiliary information
To ensure to exchange；Second process is input auxiliary information and initial ciphertext, exports a new ciphertext, initial ciphertext and new ciphertext
It is to same encryption of plaintext.
Only preferred embodiments of the present invention are described above, but are not to be construed as limiting the scope of the invention.This
Invention is not only limited to above example, and concrete structure is allowed to vary.In short, all guarantors in independent claims of the present invention
Various change is within the scope of the invention made by shield range.
Claims (3)
1. a kind of full homomorphic cryptography method of NTRU types on ring LWE, it is characterised in that：Include the following steps：
One, defined parameters：
For integer q, definitionEncryption beAnd RqIt is carried out on=R/qR,
Wherein φ (x)=xn+ 1 is cyclotomic polynomial, and n is 2 power side, q be prime number and
If polynomial f ∈ R and satisfaction | | f | |∞≤ B, then it is the boundaries B to claim f；
{χn, n ∈ N are the distributed collections on a R, if for arbitrary f ← χnHave | | f | |∞≤ B, then claim { χnIt is the boundaries B
It is distributed, i.e. the boundaries B distribution one boundaries B multinomial of output on a R；
Gaussian ProfileIt has the following properties that：ForArbitrary real numberHave：
RingThe product of upper element has following property：||s·t(modφ(x))||∞≤n·||s||∞·||t||∞；χ is on R
The boundaries B are distributed, and s1... sk← χ, then haveIt is nk-1BkBoundary；
Two, the ciphering process of a part homomorphism is carried out；
(1) parameter is established：Select μ mould q and n=(λ, μ) and Gaussian Profile χ=χ (λ, μ) so that these parameters can
Guarantee obtains 2 on ring LWEλSafety；It enablesParameter params=(q, n, χ)；
(2) generation of key：F ' ← χ is chosen, calculates f ← 2f '+1 so that f ≡ 1 (mod2)；If f is in RqIn be it is irreversible, then
Again f ', setting private key sk=f ∈ R are chosen；
(3) generation of public key：Choose g ← χ, setting public key pk=h=2gf-1∈Rq；
(4) it encrypts：Choose s, e ← χ, output ciphertext c ← hs+2e+m ∈ RqEven if public key adds an information m to obtain ciphertext c；
(5) it decrypts：Calculate μ ← fc ∈ Rq；Export m ← μm od 2；
Due to fc=fhs+2fe+fm=2gs+2fe+fm, if | | fc | |∞＜ q/2, then μ=fc, μ (mod 2)=fm (mod 2)
=m, as long as so meeting | | fc | |∞＜ q/2, then above-mentioned steps are correct；
Three, the full homomorphic cryptography process of startup is proceeded without；
(1) parameter is established：L is the number of plies of circuit, enables μ=μ (λ, L)=θ (log λ+logL), to j=0 ..., L is successively decreased
Mode sequence q0..., qL, the identical Gaussian Profile χ of each layer of use and ringParameter paramsj=
{qj, λ, χ, n }, j=0 ..., L；
(2) generation of public key：Key f is generated in the method for the second step of step 2j, in the method for the third step of step 2
Generate public key hj；It enablesIt is rightAnd fj+1Key exchange is carried out, is enabledKey sk includes fj, public key pk
Including hjWithWherein j=0 ..., L does not have ζ, each layer of circuit to have corresponding public key and private key ternary as j=L
Group
(3) it encrypts：An information m is added to be encrypted public key pk；
(4) it decrypts；Assuming that ciphertext c is in jth layer circuit, corresponding key fj, private key sk is decrypted to obtain plaintext m,
(5) additive process：Assuming that c1、c2Decruption key be fj, if not carrying out key exchange if same layer circuit；It enablesEnable f 'j=fj 2, c3Key be set as f 'j, then the method by about subtracting ciphertext noise obtains new close
Literary c4；
(6) multiplication process：Assuming that c1、c2Decruption key be fj, if not carrying out key exchange if same layer circuit；It enablesEnable f 'j=fj 2, c3Key be set as f 'j, then the method by about subtracting ciphertext noise, it obtains new
Ciphertext c4。
2. the full homomorphic cryptography method of NTRU types on ring LWE according to claim 1, it is characterised in that：Described about subtracts
The method of ciphertext noise includes two steps：The first step is that key exchanges, and ciphertext is converted into the ciphertext of next layer of circuit, key is by fj 2
It is transformed into fj+1, c1Corresponding key is fj+1, corresponding mould is qj, second step carry out mould exchange, about subtract the noise of ciphertext, c2It is right
The key answered is fj+1, corresponding mould is qj+1；fj+1Value is smaller, is selected from Gaussian Profile χ.
3. the full homomorphic cryptography method of NTRU types on ring LWE according to claim 2, it is characterised in that：The key
It includes two processes to exchange：First process is two keys of input and mould, and output auxiliary information is to ensure to exchange；Second mistake
Journey is input auxiliary information and initial ciphertext, exports a new ciphertext, initial ciphertext and new ciphertext are to same encryption of plaintext.
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| EP3304803A4 (en) * | 2015-06-02 | 2018-05-02 | Telefonaktiebolaget LM Ericsson (publ) | Method and encryption node for encrypting message |
| CN105391554B (en) * | 2015-11-09 | 2019-02-01 | 中国电子科技集团公司第三十研究所 | A kind of method and system for realizing fingerprint matching using ciphertext |
| CN106375094A (en) * | 2016-08-31 | 2017-02-01 | 江苏高网信息科技有限公司 | NTRUMLS signature method applicable to mobile payment |
| CN106533684B (en) * | 2016-12-02 | 2019-03-08 | 江苏理工学院 | Public key encryption method based on LPN |
| CN107707608A (en) * | 2017-07-26 | 2018-02-16 | 日照职业技术学院 | A kind of household Internet of Things network control system |
| CN107682140A (en) * | 2017-11-20 | 2018-02-09 | 中国科学院重庆绿色智能技术研究院 | The file encryption-decryption method of the anti-quantum attack for the low thermal expansion that multinomial point represents |
| CN109936435B (en) * | 2019-01-24 | 2022-08-30 | 中国人民武装警察部队工程大学 | NTRU type multi-key fully homomorphic encryption method with fast homomorphic operation process |
| CN109889320B (en) * | 2019-01-24 | 2022-04-15 | 中国人民武装警察部队工程大学 | BGV type multi-key fully homomorphic encryption method |
| CN110324135B (en) * | 2019-07-04 | 2022-05-31 | 浙江理工大学 | Homomorphic encryption matrix determinant security outsourcing method based on cloud computing |
| CN110545179A (en) * | 2019-08-29 | 2019-12-06 | 中芯昊月(深圳)科技控股有限公司 | R-LWE-based NTRU encryption method and security proving method thereof |
| CN111064558B (en) * | 2020-01-09 | 2023-04-07 | 浙江理工大学 | Homomorphic encryption matrix continuous multiplication secure outsourcing method based on cloud computing |
| CN111586685B (en) * | 2020-04-26 | 2022-05-03 | 重庆邮电大学 | Anonymous roaming authentication method based on lattices |
| CN112329519B (en) * | 2020-09-21 | 2024-01-02 | 中国人民武装警察部队工程大学 | Safe online fingerprint matching method |
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