CN104184581B - A kind of public key encryption method in tape symbol quadratic residue monoid - Google Patents

Abstract

The invention discloses a public key encryption method in a signed quadratic residual group, comprising the following steps: 1. Setting system parameters; 2. Generating a key; 3. The sender selects a random number to encrypt the message, And generate a ciphertext data packet; 4. The sender sends the ciphertext data packet to the receiver through the open channel; 5. After receiving the ciphertext data packet, the receiver verifies the validity of the ciphertext and uses the private key to verify the ciphertext data Decrypt the packet and get the message. The present invention simultaneously satisfies the security of confidentiality, integrity and chosen ciphertext attack, and realizes the public verifiability of ciphertext legality; because it does not use time-consuming bilinear pairing and one-time signature, it overcomes other public key encryption methods In this method, bilinear pairing or one-time signature is used to verify the legality of ciphertext, so it has higher computational efficiency and practical value.

Description

A Public Key Encryption Method in Signed Quadratic Residue Groups

technical field

The invention belongs to the field of data authentication and security in computer network communication, and in particular relates to a public key encryption method in a signed quadratic residual group.

Background technique

Cryptography is the most important means to ensure information security, and encryption is the most important component of cryptography. An encryption method involves two parties: the sender and the receiver. When the sender wants to transmit a message (or plaintext), he first encrypts it to form a ciphertext, and then sends the ciphertext to the receiver. After receiving the ciphertext, the decryptor uses the decryption algorithm to recover the original plaintext from the ciphertext. The purpose of encryption is to allow a sender and receiver to communicate secretly over an insecure channel, which may be under the control of an attacker. The attacker could be eavesdropping on, or even altering, what is being communicated between the sender and receiver. In order to ensure that the attacker cannot obtain the plaintext information in the ciphertext, the encryption method needs to satisfy certain security under a certain security model. A widely accepted standard security concept is security against chosen ciphertext attacks under the standard model. It means that in addition to the target ciphertext, the attacker can choose the ciphertext to ask the decryption machine and obtain the decryption according to his own needs, but the target ciphertext will still not reveal any information about the plaintext. An encryption method that satisfies the security against chosen ciphertext attacks under the standard model needs to verify the legitimacy of the ciphertext before decryption. When verifying the validity of the ciphertext, some methods require the private key of the decrypting party to complete, and some methods can be completed without the private key of the decrypting party. The encryption method that can complete the verification of the validity of the ciphertext without the private key of the decrypting party is called a publicly verifiable encryption method that is secure against chosen ciphertext attacks. It has important applications in firewall technology and security gateway technology.

Under the existing standard model, the secure encryption methods against chosen ciphertext attack can be divided into three categories:

The first category is constructed based on smooth hash proof systems. The structure of this type of method is relatively simple and efficient, but it does not satisfy public verifiability.

The second category is obtained by using identity-based password conversion. Such methods are publicly verifiable, but they all use time-consuming bilinear pairing techniques, so they are very inefficient.

The third category of methods is constructed based on missing trapdoor functions. These methods are publicly verifiable, but they all use inefficient one-time signature technology, so the efficiency is very low.

Contents of the invention

In view of the above defects or deficiencies, the purpose of the present invention is to provide a public key encryption method in the signed quadratic residual group, which can make full use of the special algebraic structure of the signed quadratic residual group, and design a public verification method for the legitimacy of the ciphertext , improving the efficiency of publicly verifiable public-key encryption methods.

For achieving above object, technical method of the present invention is:

Include the following steps:

Step 1: The receiver or the secure communication platform sets system parameters, and broadcasts the system parameters to all users; the system parameters include security parameters and the bit length of the hash function value;

Step 2: The recipient generates a key according to system parameters, and the key includes a public key, a verification public key, and a private key;

Step 3: The sender selects a random number, encrypts the message according to the random number and the public key, and generates a ciphertext packet;

Step 4: The sender sends the ciphertext packet to the receiver through an open channel;

Step 5: The receiver publicly verifies the validity of the ciphertext by verifying the public key, and decrypts the ciphertext with its own private key to obtain the message.

The second step specifically includes:

2.1. The receiver selects two prime numbers P and Q according to the security parameter k, and calculates N=PQ, where P=2p+1, Q=2q+1, and both p and q are prime numbers;

Choose a collision-resistant hash function H that satisfies H: Among them, Z _N is the residual group modulo N, is the signed quadratic residue group modulo N;

2.3. Random selection Two generators g ₁ , g ₂ , two exponents a, b∈[(N-1)/4];

2.4. Calculate the public key Calculate and verify the private key projection e=(b-1/2 ^v )mod pq, where The output public key is pk=(N,g ₁ ,g ₂ ,X ₁ ,X ₂ ,H), the verification public key vk=e, and the private key sk=(a,b).

Described step three comprises the following steps:

1) The sender chooses a random number y∈[(N-1)/4], and calculates

t=H(C ₁ ,C ₂ ,C ₃ )∈(1,…,2 ^v -1),

Where N=PQ=(2p+1)(2q+1), and P, Q, p, q are all prime numbers, [(N-1)/4] represents a set of integers in Indicates that (N-1)/4 is taken as an integer, m∈Z _N is the message to be sent, v is the bit length of the hash function value, is the signed quadratic residue group modulo n, X ₁ and X ₂ are receiver public keys;

2) The sender synthesizes the calculated ciphertext into an overall ciphertext packet C=(C ₁ , C ₂ , C ₃ , C ₄ ).

Described step five comprises the following steps:

1) Verify Whether it is true, if it is not true, it means that the ciphertext is illegal, and the terminator ⊥ is output;

2) Calculate t=H(C ₁ ,C ₂ ,C ₃ ), verify the equation Whether it is true, if it is not true, it means that the ciphertext is illegal, and the terminator ⊥ is output;

3) calculate m is the message to be received.

The invention provides a public key encryption method in the signed quadratic residual group, which satisfies the security of confidentiality, integrity and chosen ciphertext attack at the same time. The present invention realizes public verification of the legality of ciphertext by generating a verification public key; the present invention does not use time-consuming bilinear pairing and one-time signature, and overcomes other public key encryption methods that use bilinear pairing or one-time signature verification encryption. Therefore, it has higher computational efficiency and practical value.

Further, the present invention adds a publicly verifiable public key encryption method that is safe against chosen ciphertext attacks under the standard model to the signed quadratic residual group; the squares of two elements in the signed quadratic residual group are equivalent to The ciphertext verification method is designed based on the property that two elements are equal, which reduces the number and calculation of public and private keys to verify the validity of the ciphertext; use the signed quadratic residual group to calculate the square root equivalence of the elements when the order of the group is unknown Due to the difficult problem of factorization, the verification private key is mapped to the verification public key, and the public verification of the legality of the ciphertext is realized; the present invention does not use time-consuming bilinear pairing and one-time signature, and overcomes other public key encryption methods using dual Linear pair or one-time signature verifies the flaws of ciphertext legality, so it has higher computational efficiency and practical value.

Description of drawings

Fig. 1 is a flowchart of the present invention.

detailed description

The present invention will be described in detail below in conjunction with the accompanying drawings.

The invention provides a public key encryption method in a signed quadratic residual group, comprising the following steps:

Step 1: Set system parameters;

The receiver or a unified security communication platform is responsible for setting the security parameters. After the parameters are set, it is broadcast to all users; the system parameters to be set are as follows: k is a security parameter, and v is the bit length of the hash function value.

Step 2: Generate a key;

2.1. The receiver selects two prime numbers P and Q according to the security parameter k, and calculates N=PQ, where P=2p+1, Q=2q+1, and both p and q are prime numbers;

2.2. Select a collision-resistant hash function H to satisfy H: Among them, Z _N is the residual group modulo N, is the signed quadratic residue group modulo N;

2.3. Random selection Two generators g ₁ , g ₂ , two exponents a, b∈[(N-1)/4];

2.4. Calculate the public key Calculate and verify the private key projection e=(b-1/2 ^v )mod pq, where The output public key is pk=(N,g ₁ ,g ₂ ,X ₁ ,X ₂ ,H), the verification public key vk=e, and the private key sk=(a,b).

Step 3: The sender encrypts the message m, the specific steps are as follows:

1) The sender chooses a random number y∈[(N-1)/4], and calculates

t=H(C ₁ ,C ₂ ,C ₃ )∈(1,…,2 ^v -1),

Where N=PQ=(2p+1)(2q+1), and P, Q, p, q are all prime numbers, [(N-1)/4] represents a set of integers in Indicates that (N-1)/4 is taken as an integer, m∈Z _N is the message to be sent, v is the bit length of the hash function value, is the signed quadratic residue group modulo n, X ₁ and X ₂ are receiver public keys;

In the present invention, using the signed quadratic residual group, the squares of two elements are equal to being equal to the two elements, and a ciphertext legality verification method is designed.

2) The sender synthesizes the calculated ciphertext into an overall text packet C=(C ₁ , C ₂ , C ₃ , C ₄ ).

Step 4: The sender sends the ciphertext packet C to the receiver through an open channel;

Step 5: The recipient or any third party publicly verifies the validity of the ciphertext by verifying the public key, and the recipient decrypts the ciphertext with its own private key to obtain the message.

Specific steps are as follows:

1) Verify Whether it is true, if it is not true, it means that the ciphertext is illegal, and the terminator ⊥ is output;

2) Calculate t=H(C ₁ ,C ₂ ,C ₃ ), verify the equation Whether it is true, if it is not true, it means that the ciphertext is illegal, and the terminator ⊥ is output.

3) calculate m is the message to be received.

In step 5, using the signed quadratic residual group, when the order of the group is unknown, the square root of the calculation element is equivalent to the difficult problem of factorization, and the verification private key is mapped to the verification public key, so as to realize the reliability of the ciphertext legality Public verification.

The invention is a safe public key encryption method against chosen ciphertext attack under the standard model. When an attacker uses a forged ciphertext to ask for a secret, since in the signed quadratic residue class, the square equality of two elements means that the two elements are equal, by verifying the equality It can effectively check out illegal ciphertext, so as to resist chosen ciphertext attack.

The present invention maps the verification private key b used to verify the legitimacy of the ciphertext to obtain e=(b-1/2 ^v ) modpq, since pq is the private information of the receiver, it is obviously impossible to obtain it through e when pq is unknown b, so the mapping is a one-way trapdoor mapping in the signed quadratic residue group, and any third party can verify the legality of the ciphertext by verifying the public key e, thus realizing the public verifiability of the ciphertext legality.

The execution efficiency of the present invention specifically includes two aspects of encryption calculation and decryption calculation. Encryption uses 4 exponentiation operations, and decryption uses 3 exponentiation operations. Compared with other similar methods, the design of the present invention is simple and efficient. Specifically, assuming an exponent with a length of l bits, the exponentiation operation is equivalent to a multiplication operation with a length of 1.5l bits, the length of N l _N = 1024 bits, and the length of the hash function v = 80 bits, then encryption needs to be performed A multiplication operation with a length of 4.5v+6l _N =6504 bits, and a multiplication operation with a length of 6v+4.5l _N =5088 bits is required for decryption.

Claims (3)

1. a kind of public key encryption method in tape symbol quadratic residue monoid, it is characterised in that comprise the following steps：

Step one：Recipient or Secure Communication Environment initialization system parameter, and the systematic parameter is broadcasted to all users；Institute State bit length of the systematic parameter including security parameter and hash function value；

Step 2：Recipient includes public key, verification public key and private key according to systematic parameter generation key, the key, and public Open public key and verification public key, secrecy private key；

The step 2 is specifically included：

2.1st, recipient selects two prime Ps and Q according to security parameter k, calculates N=PQ, wherein, P=2p+1, Q=2q+1, and And p and q are prime numbers；

2.2nd, the hash function H of an anti-collision is selected, is metWherein, Z_NIt is the residue of mould N Monoid,It is the quadratic residue monoid of mould N tape symbols；

2.3rd, randomly chooseTwo generation unit g₁,g₂, two index a, b ∈ [(N-1)/4]；

2.4th, public key is calculatedCalculate checking private key Projection e=(b-1/2^v) mod pq, whereinOutput public key is pk=(N, g₁,g₂,X₁,X₂, H), checking Public key vk=e, private key sk=(a, b)；

Step 3：A sender-selected random number, and ciphertext data are generated to message encryption according to random number and public key Bag；

Step 4：Ciphertext packet is sent to recipient by sender by overt channel；

Step 5：Recipient or any third party disclose the validity of checking ciphertext by verification public key, and recipient uses oneself Private key ciphertext is decrypted, obtain message.

2. the public key encryption method in tape symbol quadratic residue monoid according to claim 1, it is characterised in that：The step Rapid three comprise the following steps：

1) sender-selected random number y ∈ [(N-1)/4], and calculate

T=H (C₁,C₂,C₃)∈(1,…,2^v-1),

Wherein N=PQ=(2p+1) (2q+1), and P, Q, p, q are prime numbers, [(N-1)/4] represent integer setWhereinExpression takes upper integer, m ∈ Z to (N-1)/4_NIt is disappearing for being sent Breath, v is the bit length of hash function value,It is the tape symbol quadratic residue monoid of mould n, X₁And X₂It is recipient's public key；

2) sender will calculate one totality ciphertext packet C=(C of gained ciphertext synthesis₁,C₂,C₃,C₄)。

3. the public key encryption method in tape symbol quadratic residue monoid according to claim 1, it is characterised in that：The step Rapid five comprise the following steps：

1) verifyWhether set up, represent that ciphertext is illegal if invalid, Output termination accords with ⊥；

2) t=H (C are calculated₁,C₂,C₃), verify equationWhether set up, the table if invalid Show that ciphertext is illegal, output termination symbol ⊥；

3) calculateM is the message to be received.