CN109768864A - Encryption method based on ECC and homomorphic cryptography - Google Patents
Encryption method based on ECC and homomorphic cryptography Download PDFInfo
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Abstract
Encryption method based on ECC and homomorphic cryptography, belongs to field of information encryption.It is oversized in order to solve public key in existing field of encryption, computation complexity is high, the low problem of operation efficiency, public key is generated using different private keys and constructs multiplicative homomorphic encryption method, the point set in insertion elliptic curve is encrypted by public key and uploads ciphertext to cloud, effect is that the encryption and decryption efficiency of inventive algorithm has apparent advantage with respect to RSA homomorphic encryption algorithm.
Description
Technical field
The invention belongs to field of information encryption, are related to a kind of encryption method based on ECC and homomorphic cryptography.
Background technique
With the development of internet and cloud computing, people search in personal information, mobile code and in terms of
Demand increasingly increase, homomorphic cryptography becomes more important.Rivest in 1978 etc. is put forward for the first time rsa encryption scheme, using same
The property of state encryption encrypt to data, and homomorphic cryptography technology has obtained further development later, achieves some generations
The research achievement of table, Peng Changgen etc. propose the general Transitive signatures scheme based on homomorphic cryptography system, are close based on homomorphism
Code system constructs safe and reliable Transitive signatures scheme and provides a kind of general framework.The it is proposeds such as Yang Yulong prevent SQL injection
The homomorphic cryptography solution of attack, Duan Shumin etc. propose the homomorphic encryption scheme based on RSA proxy re-encryption, and Yang Pan etc. is proposed
Support the CESIL homomorphic encryption scheme of ciphertext arithmetic.But there are public key size mistakes for these traditional homomorphic cryptography methods
Greatly, the disadvantages such as computation complexity is high, and operation efficiency is low, cannot be applied in cloud.
Summary of the invention
Oversized in order to solve public key in existing field of encryption, computation complexity is high, the low problem of operation efficiency, this hair
It is bright to propose a kind of encryption method based on ECC and homomorphic cryptography, it is excellent with efficient operation efficiency and high safety intensity
Point, in order to achieve the above object, technical scheme is as follows: a kind of 1. encryption methods based on ECC and homomorphic cryptography,
It is characterized in that: generating public key using different private keys and construct multiplicative homomorphic encryption method, by public key to insertion elliptic curve
In point set encrypt and upload ciphertext to cloud.
Further, user needs to store arrives cloud in plain text, and executes following steps in encryption and decryption:
1) user generates elliptic curve in the local client of cloud platform, chooses a little on it as basic point, selection is not
Same private key is to generate public key；
2) basic point and private key are carried out scalar multiplication operation to generate public key by user, save private key to locally；
3) user obtains plaintext point by plaintext embedment into the elliptic curve chosen, and carries out cryptographic operation to plaintext；
4) user generates integer at random, is then put by public key, random number and basic point to plaintext and carries out cryptographic calculation, obtained
Encrypted ciphertext；
5) user only sends ciphertext by the local client of cloud platform and stores into Cloud Server, as user needs to obtain
The product of plaintext directly calculates the product of ciphertext using multiplicative homomorphic then without decrypting the ciphertext in Cloud Server beyond the clouds
And it is sent to user；
6) user is decrypted using the private key being locally stored, and what user's decryption ciphertext obtained is built-in bright on elliptic curve
Literary dot-product, decoding obtain plaintext point.
Further, if user A needs to store (M in plain text1,M2,…,Mn) cloud is arrived, wherein n is plaintext number to be encrypted,
And following steps are executed in encryption:
1) user A generates elliptic curve E in the local client of cloud platform, chooses a point G as basic point on it, selection
Different private key (k1,k2,…,kn) to generate public key；
2) user A is by basic point G and private key (k1,k2,…,kn) scalar multiplication operation is carried out to generate public key (Q1,Q2,…,Qn),
Wherein Qi=Gki, save private key (k1,k2,…,kn) to locally；
3) user A obtains plaintext point by plaintext embedment into the elliptic curve E chosenWherein mnFor
N-th of plaintext to be encrypted, to (M in plain text1,M2,…,Mn) carry out cryptographic operation；
4) user A generates integer (r at random1,r2,…,rn), wherein random number r < n, n are the order of basic point G, are then led to
Cross public key (Q1,Q2,…,Qn), random number (r1,r2,…,rn) with basic point G to plaintext pointCarry out encryption fortune
It calculates:
I is i-th of ciphertext in formula
Encrypted ciphertext are as follows:
Further, following steps are executed in decryption:
5) user A only sends ciphertext (C by the local client of cloud platform1,C2,…,Cn) stored into Cloud Server,
As user A needs to obtain the product M of plaintext1·M2…Mn, then without decrypting ciphertext in Cloud Server, by formula (7)~(9),
It directly calculates the product of ciphertext beyond the clouds using multiplicative homomorphic and is sent to user A, product is
Wherein:
6) user A utilizes the private key (k being locally stored1,k2,…,kn) decryption:
Then:
The plaintext dot-product being built-in on elliptic curve that user A decryption ciphertext obtainsBy plaintext point decoding
For M1·M2…Mn。
The utility model has the advantages that saving space based on elliptic curve cryptography technology homomorphic cryptography method with the increase of key length
It is with the obvious advantage, and with the raising of encryption intensity, encryption method key length variation less, thus is based on elliptic curve
The advantage that the homomorphic cryptography method key of encryption technology is short is obviously.The encryption and decryption efficiency of inventive algorithm is with respect to RSA
Homomorphic encryption algorithm has apparent advantage, and RSA cryptographic algorithms cause key to generate due to being limited by prime number generation technology
It is pretty troublesome, therefore the operation efficiency of RSA homomorphic cryptography is very low.And using in the key pair present invention of different length in the present invention
Method is tested, and encrypting and decrypting efficiency is ideal.
Detailed description of the invention
Fig. 1 data ciphering and deciphering basic flow chart；
Fig. 2 point adds geometric representation figure；
Fig. 3 elliptic curve encryption processes figure；
Fig. 4 elliptic curve scatter plot.
Specific embodiment
1 homomorphic cryptography technology
Homomorphic cryptography allows server to do the operation of encryption data in the case where not knowing original plaintext, allows to encryption
Data afterwards execute specific mathematical operation and the result after decrypted result and the corresponding operation of progress in plain text is consistent, thus
Protect data.Data ciphering and deciphering basic procedure is as shown in Figure 1.
Assuming that Encryption Algorithm is CK, decipherment algorithm DK in an encryption system, it is in plain text n, m, then homomorphic cryptography meets
It is such as properties:
DK (CK (n) * CK (m))=n*m (1)
DK (CK (n)+CK (m))=n+m (2)
First attribute of above-mentioned encryption method is multiplicative homomorphic encryption, and second attribute is additive homomorphism encryption, if two
A attribute belongs to satisfaction, then Encryption Algorithm becomes complete homomorphic cryptography.Since additive homomorphism cryptographic calculation is less, relative to multiplication
Negligible, so the present invention only studies multiplicative homomorphic encryption, the research for subsequent further searching ciphertext technology provides skill
Art support.
2 elliptic curve cryptography technologies
Elliptic curve cryptography is that a kind of safety is established on the difficulty (ECDLP) for seeking Elliptic Curve Discrete Logarithm solution
Encryption method, can be described as the two o'clock G and Q on given an elliptic curve E and curve, Q=xG=G+G+...+G has altogether
The problem of x G is added, solves x is exactly the Solve problems of Elliptic Curve Discrete Logarithm.
A basic domain is let f be, x and y belong to F satisfaction: y2+ axy+by=x3+cx2+ dx+e adds an infinity
Point O just constitutes an elliptic curve.The elliptic curve equation on finite field Fp is generallyd use in general Encryption Algorithm,
It indicates are as follows:
E:y2=x3+ax+b
In formula: a, b, x, y belong to Fp, and p is a Big prime greater than 3.Assuming that the two o'clock P (x on curve1,y1), Q (x2,
y2), connect slope Δ=(y of their straight line L2-y1)/(x2-x1).L just intersects at another point R (x with elliptic curve3,y3),
Then R is the negative member of the sum of Q and P, i.e. P+Q=-R.Wherein:
x3=Δ2-x1-x2 (3)
y3=-x1+Δ(x1-x3) (4)
Point plus geometric representation such as Fig. 2.
For twice for calculating point P, crosses P one tangent line of strokes and dots and find out another point of intersection S (x3,y3), then P+P=2P=-S,
In:
Elliptic curve cryptography technology is a up-and-coming encryption system, need to will be on plaintext embedment elliptic curve when encryption
It is random some when decryption, needs the point on elliptic curve to be decoded as cleartext information, encrypting and decrypting with public key Q and private key k into
Row.Elliptic curve encryption processes such as Fig. 3.
The 3 homomorphic cryptography innovatory algorithms based on ECC
Homomorphic cryptography innovatory algorithm proposed by the present invention based on elliptic curve cryptography technology, has taken into account oval song well
The high operation efficiency of line encryption, high degree of safety and homomorphic cryptography ciphertext can operation the advantages of.The usual encryption data used time is only several
Seconds even several milliseconds, can key storage beyond the clouds can several years even decades, attacker has sufficient time to after taking ciphertext
It cracks.In addition, if one private key is cracked, and all ciphertexts will be complete in attacker's hand using same private key k encrypting plaintext
Portion is cracked, and consequence is hardly imaginable.For this purpose, the invention proposes use different privates in order to reinforce the safety of overall data
Key k is to the new thought encrypted in plain text, it is intended to further strengthen the homomorphic encryption algorithm based on elliptic curve cryptography technology
Safety.
If user A needs to store (M in plain text1,M2,…,Mn) cloud is arrived, encryption process may be summarized as follows:
1) user A generates elliptic curve E in the local client of cloud platform, chooses a point G as basic point on it, in order to
The safety for reinforcing overall data, selects different private key (k1,k2,…,kn) to generate public key.
2) user A is by basic point G and private key (k1,k2,…,kn) scalar multiplication operation is carried out to generate public key (Q1,Q2,…,Qn),
Wherein Qi=Gki, save private key (k1,k2,…,kn) to locally.
3) in order to (M in plain text1,M2,…,Mn) cryptographic operation is carried out, user A is bent to the ellipse chosen by plaintext embedment
In line E, plaintext point is obtained
4) user A generates integer (r at random1,r2,…,rn), wherein random number r < n, n are the order of basic point G, are then led to
Cross public key (Q1,Q2,…,Qn), random number (r1,r2,…,rn) with basic point G to plaintext pointCarry out encryption fortune
It calculates:
Encrypted ciphertext are as follows:
5) user A need to only send ciphertext (C by the local client of cloud platform1,C2,…,Cn) deposited into Cloud Server
Storage, as user A needs to obtain the product M of plaintext1·M2…Mn, then without decrypting ciphertext in Cloud Server, by (7)~
(9), the product of ciphertext can directly be calculated beyond the clouds using multiplicative homomorphicAnd it is sent to user A, wherein
6) user A utilizes the private key (k being locally stored1,k2,…,kn) decryption:
Then:
The plaintext dot-product being built-in on elliptic curve obtained due to user A decryption ciphertextSo last
Plaintext point is decoded as M1·M2…Mn。
Cloud only has received encrypted ciphertext, without other data availables, to prevent private message internal beyond the clouds
Leakage, while using different private key k to plaintext encrypt, greatly strengthen security intensity.Therefore, inventive algorithm is existing
The Computationally efficient of elliptic curve cryptography technique algorithm, high safety intensity, also there is isomorphism.
4 experimental results and analysis
In order to test the calculated performance of inventive algorithm, experiment porch uses Windows10, i5 processor, and client makes
Encryption and decryption is carried out with private information of the JDK1.8 to user and ciphertext is stored in Tencent's cloud platform.
4.1 safety analysis
The complexity that discrete logarithm solution is solved in the safety of elliptic curve cryptography technology and elliptic curve group is closely related,
And the difficulty for solving Elliptic Curve Discrete Logarithm solution is bigger than the RSA based on big Factorization problem, so based on oval bent
The homomorphic cryptography of line encryption technology has higher safety relative to RSA homomorphic cryptography.Such as curve y2=x3+ x+1, P=
23, scattering point is distributed as shown in figure 4, the scattering point including infinite point up to 28, presents stronger randomness
With discreteness.More than practical engineering application up to 200 points, so the Encryption Algorithm mentioned in text is enough to ensure that practical application adds
The validity of close algorithm.
Relative to general cipher mode, using homomorphic cryptography can the direct operation in ciphertext, that is, ensure that data safety
And data-privacy, and improve ciphertext transmission rate.Since the keys sizes and system parameter of elliptic curve cryptography technology are opposite
In RSA, DSA is much smaller, so memory space shared by ECC is much smaller.The security intensity of elliptic curve cryptography technology is opposite
There is obviously advantage in RSA, DSA, as shown in Table 1, the elliptic curve cryptography technique algorithm security intensity of 160 keys is just
It is equivalent to the RSA/DSA algorithm security intensity of 1024 keys, efficiently solves that increase key because improving security intensity long
The problem of Project Realization hardly possible brought by spending.Using different private key k to the mode of plaintext encryption, the safety of this method is made to exist
It is greatly improved again on the basis of the safety of elliptic curve cryptography technology.So the homomorphic cryptography method based on ECC is opposite
There are higher safety and encryption performance in RSA/DSA homomorphic cryptography method.
Key length needed for the identical security level ECC/RSA/DSA of table 1
The analysis of 4.2 calculated performances
In order to meet the demand for security of Practical Project, experiment uses the finite field P of 256bit, while using different length
The progress encrypting and decrypting time is shown in Table 2 to key pair in plain text.
2 encryptions of table/decryption time
3 inventive algorithm of table and 100 average times of RSA Algorithm operation
Encrypting and decrypting based on elliptic curve cryptography technology homomorphic cryptography method is based partially on ECC, cleartext-ciphertext operation base
In homomorphic cryptography, the increase with key length is clear to by table 1,2, is based on the method saving of elliptic curve cryptography technology homomorphic cryptography
Space it is with the obvious advantage, and with the raising of encryption intensity, encryption method key length variation less, thus based on ellipse
The advantage that the homomorphic cryptography method key of curve encryption technology is short is obviously.Inventive algorithm and RSA homomorphic cryptography are calculated
The average time of method encryption and decryption phase isolog is shown in Table 3, and the encryption and decryption efficiency of inventive algorithm has with respect to RSA homomorphic encryption algorithm
Apparent advantage, RSA cryptographic algorithms cause key to generate pretty troublesome, therefore RSA is same due to being limited by prime number generation technology
The operation efficiency of state encryption is very low.And tested in the present invention using the method in the key pair present invention of different length, add
Close decryption efficiency is ideal.
5 beneficial effects
With the rapid development of cloud computing and big data, safety problem becomes urgent problem.The present invention combines
Elliptic curve cryptography technology and homomorphic cryptography technology, propose a kind of improved homomorphic cryptography method, and this method is existing oval bent
The Computationally efficient of line encryption technology algorithm, short key, high safety intensity also have the ciphertext of homomorphic cryptography can operating characteristic.Make
With different private key k to data encryption, solves a private key and be cracked, the problem of total data is all revealed, thus greatly
Improve the safety of the homomorphic cryptography based on elliptic curve cryptography technology.For current RSA (Rivest-Shamir-
Adleman the disadvantage that) homomorphic cryptography method computational efficiency is low, public key size is big, computation complexity is high, in conjunction with ECC (elliptic
Curve cryptography) and homomorphic cryptography method, propose a kind of improved homomorphic encryption algorithm.In elliptic curve cryptography
On the basis of algorithm, public key is generated using different private keys and constructs multiplicative homomorphic encryption method, it is oval to insertion by public key
Point set in curve encrypts and uploads ciphertext to cloud.Theory analysis with the experimental results showed that, relative to current homomorphic cryptography
Method, improved Encryption Algorithm solve the problems, such as that public key size is big and computation complexity is high while ensuring safety,
And it can be directly to ciphertext operation in cloud without being decrypted into plain text.
Claims (4)
1. a kind of encryption method based on ECC and homomorphic cryptography, it is characterised in that: generate public key using different private keys and construct
Multiplicative homomorphic encryption method encrypts the point set in insertion elliptic curve by public key and uploads ciphertext to cloud.
2. the encryption method based on ECC and homomorphic cryptography as described in claim 1, it is characterised in that: user needs to store bright
Text arrives cloud, and executes following steps in encryption and decryption:
1) user generates elliptic curve in the local client of cloud platform, chooses a little as basic point, selects different on it
Private key is to generate public key；
2) basic point and private key are carried out scalar multiplication operation to generate public key by user, save private key to locally；
3) user obtains plaintext point by plaintext embedment into the elliptic curve chosen, and carries out cryptographic operation to plaintext；
4) user generates integer at random, is then put by public key, random number and basic point to plaintext and carries out cryptographic calculation, is encrypted
Ciphertext afterwards；
5) user only sends ciphertext by the local client of cloud platform and stores into Cloud Server, as user needs to obtain in plain text
Product, then without decrypting ciphertext in Cloud Server, the product for directly calculating ciphertext beyond the clouds using multiplicative homomorphic is concurrent
It send to user；
6) user is decrypted using the private key being locally stored, and user decrypts the plaintext point being built-in on elliptic curve that ciphertext obtains
Product, decoding obtain plaintext point.
3. the encryption method based on ECC and homomorphic cryptography as described in claim 1, it is characterised in that: set user A and need to store
(M in plain text1,M2,···,Mn) cloud is arrived, wherein n is plaintext number to be encrypted, and following steps are executed in encryption:
1) user A generates elliptic curve E in the local client of cloud platform, chooses a point G as basic point on it, selection is different
Private key (k1,k2,···,kn) to generate public key；
2) user A is by basic point G and private key (k1,k2,···,kn) scalar multiplication operation is carried out to generate public key (Q1,Q2,···,
Qn), wherein Qi=Gki, save private key (k1,k2,···,kn) to locally；
3) user A obtains plaintext point by plaintext embedment into the elliptic curve E chosenWherein mnIt is n-th
Plaintext to be encrypted, to (M in plain text1,M2,···,Mn) carry out cryptographic operation；
4) user A generates integer (r at random1,r2,···,rn), wherein random number r < n, n are the order of basic point G, are then led to
Cross public key (Q1,Q2,···,Qn), random number (r1,r2,···,rn) with basic point G to plaintext pointIt carries out
Cryptographic calculation:
I is i-th of ciphertext in formula
Encrypted ciphertext are as follows:
4. the encryption method based on ECC and homomorphic cryptography as claimed in claim 3, it is characterised in that: executed such as in decryption
Lower step:
5) user A only sends ciphertext (C by the local client of cloud platform1,C2,···,Cn) stored into Cloud Server,
As user A needs to obtain the product M of plaintext1·M2···Mn, then without decrypting ciphertext in Cloud Server, by formula (7)~
(9), it directly calculates the product of ciphertext beyond the clouds using multiplicative homomorphic and is sent to user A, product is
Wherein:
6) user A utilizes the private key (k being locally stored1,k2,…,kn) decryption:
Then:
The plaintext dot-product being built-in on elliptic curve that user A decryption ciphertext obtainsPlaintext point is decoded as M1·
M2···Mn。
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| CN115484172A (en) * | 2022-09-16 | 2022-12-16 | 宝吉良 | Management system and method of computer control terminal based on Internet of things |
| CN115865307A (en) * | 2023-02-27 | 2023-03-28 | 蓝象智联(杭州)科技有限公司 | Data point multiplication operation method for federal learning |
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| CN110602083A (en) * | 2019-09-10 | 2019-12-20 | 上海黔易数据科技有限公司 | Secure transmission and storage method of digital identity authentication data |
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| CN111431710A (en) * | 2020-03-24 | 2020-07-17 | 数据通信科学技术研究所 | Encryption method and device, reference method and device, supervision method and device |
| CN111431710B (en) * | 2020-03-24 | 2021-09-10 | 数据通信科学技术研究所 | Encryption method and device allowing sender to look up and third party to supervise |
| CN112769542A (en) * | 2021-04-12 | 2021-05-07 | 富算科技(上海)有限公司 | Multiplication triple generation method, device, equipment and medium based on elliptic curve |
| CN112769542B (en) * | 2021-04-12 | 2021-06-11 | 富算科技(上海)有限公司 | Multiplication triple generation method, device, equipment and medium based on elliptic curve |
| CN113141247A (en) * | 2021-04-25 | 2021-07-20 | 重庆都会信息科技有限公司 | Homomorphic encryption method, device and system and readable storage medium |
| CN113141247B (en) * | 2021-04-25 | 2023-07-04 | 重庆都会信息科技有限公司 | Homomorphic encryption method, homomorphic encryption device, homomorphic encryption system and readable storage medium |
| CN115484172A (en) * | 2022-09-16 | 2022-12-16 | 宝吉良 | Management system and method of computer control terminal based on Internet of things |
| CN115484172B (en) * | 2022-09-16 | 2023-09-26 | 深圳市兴海物联科技有限公司 | Management system and method of computer control terminal based on Internet of things |
| CN115865307A (en) * | 2023-02-27 | 2023-03-28 | 蓝象智联(杭州)科技有限公司 | Data point multiplication operation method for federal learning |
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