CN115567308A - A Data Encryption Transmission Method Based on Key Expansion - Google Patents

Abstract

The invention relates to a data encryption transmission method based on key expansion, which comprises the steps of firstly executing a key sequence expansion method according to preset various key expansion methods to sequentially obtain keys of each round of rounds; sequentially cutting the plaintext to be transmitted to obtain independent plaintext blocks with the same length as the initial secret key; then, sequentially applying each round key to encrypt the independent plaintext blocks to obtain each independent ciphertext block, encrypting each round key and related preset information to obtain a key encodedKey information, and sending each encrypted data to a receiver by a sender; finally, the receiver decrypts the plaintext to complete the encrypted transmission of the plaintext to be transmitted from the sender to the receiver; the design adopts a key expansion mode, carries out a multi-stage encryption technology on the plaintext to be transmitted, and combines the encryption operation of the key, thereby not only ensuring the security of the key, but also ensuring the security and the encryption efficiency of the plaintext in the application.

Description

A Data Encryption Transmission Method Based on Key Expansion

technical field

The invention relates to a data encryption transmission method based on key extension, belonging to the technical field of data encryption.

Background technique

Cryptography is widely used and has become an indispensable security technology in people's daily life. For example, the social chat tools that people usually use involve a large amount of personal privacy information. In order to ensure the security of users' personal information and chat information, the existing technology encrypts the chat records stored locally and the information transmitted over the network; In addition to the protection of chat information and personal privacy in existing applications, payment is also one of the businesses that often use cryptography. In addition to the above two familiar tools that use cryptography, banking, e-government, satellite communications, wireless transmission, etc. all involve cryptography, so it is necessary to study cryptography.

Symmetric encryption (also known as private key encryption) algorithm, which is named after the same key is used in the encryption and decryption process, which is what we usually call the traditional (block) encryption algorithm. Since the encryption and decryption process of the algorithm requires the same key, in order to use this algorithm for communication, both parties (sender and receiver) need to negotiate to obtain a shared key that only both parties know before conducting secure communication. Used for encryption and decryption. The security of the symmetric algorithm mainly depends on the confidentiality of the key. Once the key used in the communication process is obtained by a third party, it means that the third party can use the obtained symmetric key to decrypt the sent and received ciphertext messages. Therefore, the secrecy of the key is the key to symmetric encrypted communication.

The RSA algorithm is a commonly used method for encrypting information and is the basis of Internet communication encryption. The public key and private key can be generated using the RSA algorithm. The private key is kept by the server, and the public key is kept by the client. At the same time, the private key can decrypt the content encrypted by the public key, and the information encrypted by the private key can only be decrypted by the public key. Therefore, as long as the private key is not leaked, the communication security between the two parties will be guaranteed.

Contents of the invention

The technical problem to be solved by the present invention is to provide a data encryption transmission method based on key expansion, which adopts a new design logic, applies key expansion and multi-level encryption technology, and can effectively improve the security of data transmission.

In order to solve the above-mentioned technical problems, the present invention adopts the following technical solutions: the present invention designs a data encryption transmission method based on key expansion, which is used to realize the encryption transmission of the plaintext to be transmitted from the sender to the receiver, including the following steps:

Step A. Based on the seed key composed of the preset number of words corresponding to the preset length initial key, and each word of equal length, the seed key is used as the first round of round keys, according to the preset various key expansion methods The key sequence expansion method with the preset number of rounds is composed of a preset sort, and the key expansion process is performed sequentially, and the round keys with the same length as the initial key are obtained in sequence, and then enter step B;

Step B. Segment the plaintext to be transmitted sequentially to obtain each independent plaintext block equal to the length of the initial key, wherein if the length of the last independent plaintext block is insufficient, then fill in 0 to satisfy the length; then enter step C;

Step C. For each independent plaintext block, the sender sequentially applies each round key to perform encryption processing on each independent plaintext block, obtains corresponding independent ciphertext blocks, and then obtains each independent ciphertext block, and then enters step D;

Step D. The sender uses the recipient's public key to encrypt each round key and the preset information involved in steps A to B to obtain the encrypted key encodedKey information, and the sender encrypts each The independent ciphertext block and the key encodedKey information are sent to the receiver, and then enter step E;

Step E. The recipient uses its private key to decrypt the encodedKey information from the sender, obtains the round keys and the preset information involved in steps A to B, and performs the reverse operation of step B , to decrypt each independent ciphertext block from the sender to obtain each independent plaintext block, that is, to complete the encrypted transmission of the plaintext to be transmitted from the sender to the receiver.

As a preferred technical solution of the present invention: in the step A, based on the 128-bit initial key corresponding to the 4 seed keys that are respectively 32-bit words, the seed key is used as the first-round key, According to the key sequence expansion method of the preset number of rounds formed by the preset various key expansion methods and the preset sorting, for the 4 characters in the round keys obtained in each round except the last round, according to the corresponding round The key expansion method performs key expansion and update, obtains 4 words in the next round key to form the next round key, and sequentially obtains the 128-bit round key of each round.

As a preferred technical solution of the present invention: the various key expansion methods include a key expansion method R1 and a key expansion method R2, wherein the key expansion method R1 is as follows:

The fourth word in the round key is processed according to the preset function transformation T, and the processing result is XORed with the first word in the round key to obtain the key extension corresponding to the first word in the round key updated results;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the second word in the round key and the first word in the round key to obtain the expanded and updated result of the key corresponding to the second word in the round key;

Exclusive OR processing is performed on the result after the key expansion and update corresponding to the third word in the round key and the second word in the round key to obtain the result after the key expansion and update corresponding to the third word in the round key;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the fourth word in the round key and the third word in the round key to obtain the expanded and updated result of the key corresponding to the fourth word in the round key;

Then, the four words in the next round key are formed by the key expansion and update results corresponding to the first word, the second word, the third word, and the fourth word in the round key, and the following round key;

The key expansion method R2 is as follows:

The first word in the round key is processed according to the preset function transformation T, and the processing result is XORed with the fourth word in the round key to obtain the corresponding key extension of the fourth word in the round key updated results;

Exclusive OR processing is performed on the result after the key expansion and update corresponding to the third word in the round key and the fourth word in the round key to obtain the result after the key expansion and update corresponding to the third word in the round key;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the second word in the round key and the third word in the round key to obtain the expanded and updated result of the key corresponding to the second word in the round key;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the first word in the round key and the second word in the round key to obtain the expanded and updated result of the key corresponding to the first word in the round key;

Then, the four words in the next round key are formed by the key expansion and update results corresponding to the first word, the second word, the third word, and the fourth word in the round key, and the following Round key.

As a preferred technical solution of the present invention: the preset function transformation T sequentially includes three operations of the first row shift, the first byte substitution and the first round of constant XOR, wherein the first row shift represents Move 4 bytes in 1 word circularly to the left or right by 1 byte position; the first byte substitution means that for 4 bytes in 1 word, use 1 byte The upper 4 bits are used as the row value, and the lower 4 bits are used as the column value, and the corresponding elements are taken out from the preset two-dimensional matrix S box for output, and then the output corresponding to each byte is obtained; the first round of constant XOR indicates that it corresponds to the first word The result of section substitution processing is XORed with the round constant.

As a preferred technical solution of the present invention: each round function is preset in a one-to-one correspondence based on the second round key to the last round key, and the second round key to the penultimate round key are respectively The corresponding round function includes the second byte substitution performed sequentially, the second row shift, column mixing, and the application of the corresponding round key to perform key round addition. The round function corresponding to the last round key includes the sequence Perform the second byte substitution, the second line shift, and the application of the corresponding round key to perform key round addition; in the step C, the sender targets each independent plaintext block, and firstly targets the corresponding independent plaintext block The state matrix in units of bytes, the first round key is used for key round addition processing, and then the processing results are processed by the corresponding round functions from the second round key to the last round key respectively , to obtain the corresponding independent ciphertext block;

Among them, the second byte substitution means that for each byte in the state matrix, the upper 4 bits of the byte are used as the row value and the lower 4 bits are used as the column value, and the corresponding elements are extracted from the preset two-dimensional matrix S box for output , and then obtain the output corresponding to each byte, and update the state matrix; the second row shift means that each row in the state matrix is shifted to the left or right in a single direction to move the corresponding row number bytes, and the state matrix is updated; the column mix represents the state matrix Performing multiplication processing with the preset fixed matrix to obtain the obfuscated state matrix; applying the corresponding round key to perform key round addition means applying the corresponding round key to perform key round addition processing on the state matrix.

As a preferred technical solution of the present invention: the recipient's public key and private key are obtained as follows;

The receiver first randomly selects two prime numbers p and q that are not equal to each other, and obtains N according to N=p*q;

Then calculate L according to L=lcm(p-1,q-1), where lcm(X,Y) represents the least common multiple of X and Y;

Then according to 1<E<L, and gcd(E,L)=1, calculate E, gcd(X,Y) to represent the greatest common divisor of X and Y;

Then according to 1<D<L, E*DmodL=1, calculate and obtain D, that is, obtain the recipient's public key (N, E) and the recipient's private key (N, D).

A data encryption transmission method based on key expansion according to the present invention, compared with the prior art by adopting the above technical scheme, has the following technical effects:

The present invention designs a data encryption transmission method based on key expansion. First, the seed key is used as the first round key, and the key sequence expansion method is executed according to various preset key expansion methods, and each round and key are sequentially obtained. A round key with the same length as the initial key; then divide the plaintext to be transmitted sequentially to obtain each independent plaintext block equal to the length of the initial key; then apply each round key sequentially to the independent plaintext blocks for encryption processing, Obtain each independent ciphertext block, and encrypt each round key and the preset information involved to obtain the key encodedKey information, and send each encrypted data to the receiver by the sender; finally, the receiver decrypts it, Complete the encrypted transmission of the plaintext to be transmitted from the sender to the receiver; design multi-level encryption technology for the plaintext to be transmitted through the key expansion method, combined with the encryption operation of the key, not only to ensure the security of the key, but also to apply It can guarantee the security of plaintext and the efficiency of encryption.

Description of drawings

Fig. 1 is the flow chart of the present invention's design based on the data encryption transmission method of key expansion;

Fig. 2 is a schematic diagram of the framework of the key expansion method R1 involved in the present invention;

Fig. 3 is a schematic diagram of the framework of the key expansion method R2 involved in the present invention;

Fig. 4 is a schematic diagram of the encryption processing flow for each independent plaintext block in step C designed by the present invention.

detailed description

The specific implementation manners of the present invention will be further described in detail below in conjunction with the accompanying drawings.

The present invention designs a data encryption transmission method based on key expansion, which is used to realize the encrypted transmission of the plaintext to be transmitted from the sender to the receiver. In practical applications, as shown in Figure 1, the following steps A to E are specifically performed. .

Step A. Based on the seed key composed of the preset number of words corresponding to the preset length initial key, and each word of equal length, the seed key is used as the first round of round keys, according to the preset various key expansion methods The key sequence expansion method with the preset number of rounds is composed of a preset sequence, and the key expansion process is performed in sequence, and the round keys with the same length as the initial key are sequentially obtained, and then step B is entered.

In practical applications, in the above step A, based on the seed key composed of 4 words of 32 bits corresponding to the 128-bit initial key, the seed key is used as the first round key, according to the preset various encryption keys. The key expansion method uses a preset sorting method to form a key sequence expansion method with a preset number of rounds, and sequentially targets the four characters in each round key obtained except the last round, according to the key expansion method in the corresponding round. Extend the update to obtain 4 words in the next round of round keys to form the next round of round keys, and obtain the 128-bit round keys of each round in sequence.

Among them, various key expansion methods include key expansion method R1 and key expansion method R2, that is, a key sequence expansion method with a preset number of rounds composed of R1 and R2 in a preset order, such as {R1, R2, R2, R1...R1, R2}, where, as shown in Figure 2, the key expansion method R1 is as follows:

The fourth word in the round key is processed according to the preset function transformation T, and the processing result is XORed with the first word in the round key to obtain the key extension corresponding to the first word in the round key updated results;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the second word in the round key and the first word in the round key to obtain the expanded and updated result of the key corresponding to the second word in the round key;

Exclusive OR processing is performed on the result after the key expansion and update corresponding to the third word in the round key and the second word in the round key to obtain the result after the key expansion and update corresponding to the third word in the round key;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the fourth word in the round key and the third word in the round key to obtain the expanded and updated result of the key corresponding to the fourth word in the round key;

Then, the four words in the next round key are formed by the key expansion and update results corresponding to the first word, the second word, the third word, and the fourth word in the round key, and the following Round key.

And as shown in Figure 3, the key expansion method R2 is as follows:

The first word in the round key is processed according to the preset function transformation T, and the processing result is XORed with the fourth word in the round key to obtain the corresponding key extension of the fourth word in the round key updated results;

Exclusive OR processing is performed on the result after the key expansion and update corresponding to the third word in the round key and the fourth word in the round key to obtain the result after the key expansion and update corresponding to the third word in the round key;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the second word in the round key and the third word in the round key to obtain the expanded and updated result of the key corresponding to the second word in the round key;

Exclusive OR processing is performed on the expanded and updated result of the key corresponding to the first word in the round key and the second word in the round key to obtain the expanded and updated result of the key corresponding to the first word in the round key;

Then, the four words in the next round key are formed by the key expansion and update results corresponding to the first word, the second word, the third word, and the fourth word in the round key, and the following Round key.

Here, the preset function transformation T sequentially includes three operations: the first row shift, the first byte substitution, and the first round of constant XOR, where the first row shift means to cycle 4 bytes in a word Move 1 byte position in a single direction to the left or right; the first byte substitution means that for 4 bytes in 1 word, the high 4 bits of 1 byte are used as the row value, and the low 4 bits As a column value, the corresponding element is taken out from the preset two-dimensional matrix S box for output, and then the output corresponding to each byte is obtained; the first round of constant XOR means that the result of the corresponding first byte substitution processing is the same round of constant XOR .

Step B. Segment the plaintext to be transmitted sequentially to obtain independent plaintext blocks equal to the length of the initial key. If the length of the last independent plaintext block is insufficient, fill in 0s to satisfy the length; then enter step C.

Step C. For each independent plaintext block, the sender sequentially applies each round key for encryption processing to obtain the corresponding independent ciphertext block, and then obtains each independent ciphertext block, and then enters step D.

In practical applications, as shown in FIG. 4 , the preset round functions are one-to-one based on the second round key to the last round key, and the second round key to the penultimate round key are respectively The corresponding round function includes the second byte substitution performed sequentially, the second row shift, column mixing, and the application of the corresponding round key to perform key round addition. The round function corresponding to the last round key includes the sequence Perform the second byte substitution, the second line shift, and the application of the corresponding round key to perform key round addition; in the step C, the sender targets each independent plaintext block, and firstly targets the corresponding independent plaintext block The state matrix in units of bytes, the first round key is used for key round addition processing, and then the processing results are processed by the corresponding round functions from the second round key to the last round key respectively , to obtain the corresponding independent ciphertext block.

Among them, the second byte substitution means that for each byte in the state matrix, the upper 4 bits of the byte are used as the row value and the lower 4 bits are used as the column value, and the corresponding elements are extracted from the preset two-dimensional matrix S box for output , and then obtain the output corresponding to each byte, and update the state matrix; the second row shift means that each row in the state matrix is shifted to the left or right in a single direction to move the corresponding row number bytes, and the state matrix is updated; the column mix represents the state matrix Performing multiplication processing with the preset fixed matrix to obtain the obfuscated state matrix; applying the corresponding round key to perform key round addition means applying the corresponding round key to perform key round addition processing on the state matrix.

In the application, the recipient's public key and private key are obtained as follows.

The receiver first randomly selects two prime numbers p and q that are not equal to each other, and obtains N according to N=p*q;

Then calculate L according to L=lcm(p-1,q-1), where lcm(X,Y) represents the least common multiple of X and Y;

Then according to 1<E<L, and gcd(E,L)=1, calculate E, gcd(X,Y) to represent the greatest common divisor of X and Y;

Then according to 1<D<L, E*DmodL=1, calculate and obtain D, that is, obtain the recipient's public key (N, E) and the recipient's private key (N, D).

Step D. The sender uses the recipient's public key to encrypt each round key and the preset information involved in steps A to B to obtain the encrypted key encodedKey information, and the sender encrypts each The independent ciphertext block and the key encodedKey information are sent to the receiver, and then step E is entered.

Step E. The recipient uses its private key to decrypt the encodedKey information from the sender, obtains the round keys and the preset information involved in steps A to B, and performs the reverse operation of step B , to decrypt each independent ciphertext block from the sender to obtain each independent plaintext block, that is, to complete the encrypted transmission of the plaintext to be transmitted from the sender to the receiver.

A data encryption transmission method based on key expansion is designed above. First, the seed key is used as the first round key, and the key sequence expansion method is executed according to various preset key expansion methods, and each round and the initial key are sequentially obtained. The round key with the same key length; then divide the plaintext to be transmitted sequentially to obtain each independent plaintext block with the same length as the initial key; then apply each round key to the independent plaintext block for encryption in order to obtain Each independent ciphertext block is encrypted for each round key and the preset information involved to obtain the key encodedKey information, and the sender sends each encrypted data to the receiver; finally, the receiver decrypts and completes The sender encrypts the plaintext to be transmitted to the receiver; the design adopts the key expansion method to carry out multi-level encryption technology for the plaintext to be transmitted, combined with the encryption operation of the key, not only ensures the security of the key, but also It can guarantee the security of plaintext and the efficiency of encryption.

The embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above embodiments, and can also be made without departing from the gist of the present invention within the scope of knowledge possessed by those of ordinary skill in the art. Variations.

Claims (6)

1. A data encryption transmission method based on key expansion is used for realizing encryption transmission of plaintext to be transmitted from a sender to a receiver, and is characterized by comprising the following steps:

a, based on a seed key composed of words with the preset number and the same length corresponding to an initial key with the preset length, taking the seed key as a first round key, sequentially executing key expansion processing according to a key sequence expansion method with preset rounds composed of preset sequences by various preset key expansion methods, sequentially obtaining round keys with the same length as the initial key in each round, and then entering the step B;

b, sequentially cutting the plaintext to be transmitted to obtain independent plaintext blocks with the length equal to that of the initial secret key, wherein if the length of the last independent plaintext block is insufficient, 0 is supplemented to meet the length; then entering step C;

step C, the sender applies each round of secret keys to each independent plaintext block and each independent plaintext block in sequence to carry out encryption processing to obtain corresponding independent ciphertext blocks, and then each independent ciphertext block is obtained, and then the step D is carried out;

d, encrypting each round key and preset information in the steps A to B by using a public key of a receiver by a sender to obtain encrypted key encodedKey information, sending each independent ciphertext block and the key encodedKey information to the receiver by the sender, and then entering the step E;

and E, the receiver decrypts the key encodedKey information from the sender by using the private key of the receiver to obtain the round keys and the preset information in the steps A to B, decrypts each independent ciphertext block from the sender according to the inverse operation of the step B to obtain each independent plaintext block, and namely, the sender completes the encrypted transmission of the plaintext to the receiver about to be transmitted.

2. The data encryption transmission method based on the key expansion as claimed in claim 1, characterized in that: in the step a, based on a seed key composed of 4 words of 32 bits respectively corresponding to a 128-bit initial key, the seed key is used as a first round key, and according to a key sequence expansion method of a preset round number composed of preset sequences of preset various key expansion methods, key expansion updating is performed on 4 words of keys of each round except for the last round in sequence according to a key expansion method in a corresponding round, so as to obtain 4 words of keys of a next round, thereby forming a key of the next round, and sequentially obtaining a round key of 128 bits of each round.

3. The data encryption transmission method based on the key expansion as claimed in claim 1 or 2, characterized in that: the various key expansion methods comprise a key expansion method R1 and a key expansion method R2, wherein the key expansion method R1 is as follows: processing the 4 th word in the round key according to a preset function transformation T, and performing exclusive OR processing on the processing result and the 1 st word in the round key to obtain a result of the round key after the key corresponding to the 1 st word is expanded and updated;

performing XOR processing on the result of the expanded and updated key corresponding to the 2 nd word in the round key and the 1 st word in the round key to obtain the result of the expanded and updated key corresponding to the 2 nd word in the round key;

performing XOR processing on the result of the expanded and updated key corresponding to the 3 rd word in the round key and the 2 nd word in the round key to obtain the result of the expanded and updated key corresponding to the 3 rd word in the round key;

performing XOR processing on the result of the key expansion update corresponding to the 4 th word in the round key and the 3 rd word in the round key to obtain the result of the key expansion update corresponding to the 4 th word in the round key;

then expanding and updating results of keys corresponding to the 1 st word, the 2 nd word, the 3 rd word and the 4 th word in the round keys to form 4 words in the next round keys, and obtaining the next round keys;

the key expansion method R2 is as follows:

processing the 1 st word in the round key according to a preset function transformation T, and performing XOR processing on the processing result and the 4 th word in the round key to obtain a result of the round key after the 4 th word corresponds to the key expansion update;

performing XOR processing on the result of the expansion and update of the key corresponding to the 3 rd word in the round key and the 4 th word in the round key to obtain the result of the expansion and update of the key corresponding to the 3 rd word in the round key;

performing XOR processing on the result of the expanded and updated keys corresponding to the 2 nd word and the 3 rd word in the round key to obtain the result of the expanded and updated keys corresponding to the 2 nd word in the round key;

performing XOR processing on the result of the key expansion update corresponding to the 1 st word in the round key and the 2 nd word in the round key to obtain the result of the key expansion update corresponding to the 1 st word in the round key;

and then expanding and updating the corresponding keys of the 1 st word, the 2 nd word, the 3 rd word and the 4 th word in the round key to form 4 words in the next round key, and obtaining the next round key.

4. The data encryption transmission method based on the key expansion as claimed in claim 3, characterized in that: the preset function transformation T sequentially comprises three operations of first line shifting, first byte substitution and first round constant XOR, wherein the first line shifting represents that 4 bytes in 1 word are circularly moved to a single direction of left or right by 1 byte position; the first byte substitution represents that aiming at 4 bytes in 1 word, the high 4 bits of 1 byte are used as row values and the low 4 bits are used as column values, corresponding elements are taken out by a preset two-dimensional matrix S box to be output, and then output corresponding to each byte is obtained; the first round of constant xor indicates that the result of the corresponding first byte substitution process is xor-ed with the round constant.

5. The data encryption transmission method based on the key expansion as claimed in claim 1, characterized in that: the method comprises the steps that round functions are preset on the basis of the first round key, the last round key and the corresponding round keys one by one, the round functions corresponding to the first round key and the last round key respectively comprise sequentially executed first byte substitution, first row displacement, column mixing and corresponding round key application execution key round addition, and the round functions corresponding to the last round key comprise sequentially executed first byte substitution, first row displacement and corresponding round key application execution key round addition; in the step C, the sender applies the first round key to perform key round addition processing on the state matrix which corresponds to each independent plaintext block and takes the byte as a unit, and then sequentially processes the processing result through the second round key to the last round key to obtain the corresponding independent ciphertext block;

the second byte substitution represents that aiming at each byte in the state matrix, the high 4 bits of the byte are used as row values and the low 4 bits of the byte are used as column values, corresponding elements are taken out by a preset two-dimensional matrix S box to be output, and then output corresponding to each byte is obtained, and the state matrix is updated; the second row shifting represents that the corresponding row number bytes are respectively shifted in the single direction of left or right aiming at each row in the state matrix, and the state matrix is updated; multiplying the column mixed representation state matrix by a preset fixed matrix to obtain a confused state matrix; and the key round addition executed by applying the corresponding round key represents that the key round addition processing is executed by applying the corresponding round key aiming at the state matrix.

6. The data encryption transmission method based on the key expansion as claimed in claim 1, characterized in that: the public key and the private key of the receiver are obtained as follows;

the receiver firstly randomly selects two mutually unequal prime numbers p and q, and obtains N according to N = p × q;

obtaining L through calculation according to L = lcm (p-1, q-1), wherein lcm (X, Y) represents the least common multiple of X and Y;

then E, gcd (X, Y) is calculated according to 1 Once E < -L and gcd (E, L) =1 to express the greatest common divisor of X and Y; then, according to 1-D-T and E-Dmod L =1, D is obtained through calculation, namely, the public key (N, E) of the receiver and the private key (N, D) of the receiver are obtained.

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