CN110855436A - Structure of key system based on secondary surplus - Google Patents

Structure of key system based on secondary surplus Download PDF

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CN110855436A
CN110855436A CN201911117914.5A CN201911117914A CN110855436A CN 110855436 A CN110855436 A CN 110855436A CN 201911117914 A CN201911117914 A CN 201911117914A CN 110855436 A CN110855436 A CN 110855436A
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bob
sequence
ciphertext
plaintext
mod
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罗婧
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Anhui University of Science and Technology
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Anhui University of Science and Technology
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
    • H04L9/0861—Generation of secret information including derivation or calculation of cryptographic keys or passwords

Abstract

The invention relates to a structure of a key system based on secondary residue; in order to avoid the problem that a deterministic public key cryptosystem is easy to be attacked by selecting plaintext, the invention introduces random numbers on the basis of the difficulty of large number factorization and the difficulty of square root solving by quadratic residue of a modular Blum number, and constructs a polynomial encryption key system on the basis of quadratic residue. The cipher strength of the cipher system is not lower than that of RSA public key cipher system, and this can resist attack from selected plaintext and has high safety. The expansion rate of the ciphertext is not higher than that of the BG cryptosystem proposed by Blum and Goldwasser, and when the plaintext to be transmitted is longer, the expansion rate of the ciphertext is approximate to 1.

Description

Structure of key system based on secondary surplus
Technical Field
The invention relates to the field of cryptography, in particular to a structure of a key system based on secondary residue.
Background
With the advent of the cloud era, many people upload their own important information to an internet server, and the security of internet communication and shared data is concerned. To ensure the security of data, it is the information encryption that is done first. There are many ways of encryption in cryptography, and the public key cryptosystem has an irreplaceable role due to its unique advantages. When we have chosen the encryption key, the correspondence between each plaintext and each ciphertext is bijective. However, when we encrypt plaintext, for the public key cryptosystem, the key of the encryption algorithm is usually published in advance. Therefore, it is generally difficult to withstand the attack of choosing plaintext with a certain public key cryptosystem. For the regret of the public key cryptosystem, Goldwasser and Micali put forward the concept of the probabilistic cryptosystem for the first time in 1984, but the expansion rate is too large, so that the probabilistic cryptosystem has no practical value.
Based on the thought of quadratic residue and probability encryption, the invention deeply studies the probability public key cipher and constructs a polynomial encryption key system based on quadratic residue. The newly constructed secondary residual key system can well resist the attack of selecting plaintext, and simultaneously, the expansion rate of the ciphertext of the secondary residual key system is lower than that of the BG cryptosystem proposed by Blum and Goldwasser, and the security of the cryptosystem is not lower than that of an RSA public key cryptosystem.
Disclosure of Invention
The invention aims to provide a structure of a key system based on secondary residue; has the advantages that: the cipher strength of the cipher system is not lower than that of RSA public key cipher system, and this can resist attack from selecting plain text effectively and has high safety. The expansion rate of the ciphertext is not higher than that of the BG cryptosystem proposed by Blum and Goldwasser, and when the plaintext to be transmitted is longer, the expansion rate of the ciphertext is approximate to 1.
The invention adopts the following technical scheme for realizing the purpose: the method comprises the following steps:
(1) encryption algorithm
1): at ZnThe n-th irreducible polynomial p (x) a0+a1x+a2x2+…+anxnThe plaintext M is (a)0,a1,…an)，(a0,a1,…an) A sequence of coefficients that is an irreducible polynomial of degree n.
2): alice finds the encryption key N published in advance by Bob, and calculates as follows:
Figure BDA0002274590580000011
b1≡a1 2modN (2)
Figure BDA0002274590580000021
bn≡an 2modN (3)
to obtain the sequence B ═ B: (b0,b1,b2,…bn)
3): a random variable k is introduced and the random variable k,the following calculations were made:
x0≡k2modN (4)
x1≡x0 2modN (5)
Figure BDA0002274590580000022
xn+1≡xn 2modN (6)
to give the sequence D ═ (x)1,x2,x3…xn+1)
4): then calculate xn+2≡xn+1 2modN
5): calculating E ═ b0×x1,b1×x2,…bn×xn+1) Record c1＝b0×x1，c2＝b1×x2，…，cn+1＝bn×xn+1
5): transmitting ciphertext C ═ C1,c2,…cn+1,xn+2) To Bob.
(2) Decryption algorithm
1): bob obtains the ciphertext C ═ C (C) sent by Alice1,c2,…cn+1,xn+2) Then, according to theorem 1.5, we calculate in turn:
Figure BDA0002274590580000024
Figure BDA0002274590580000025
Figure BDA0002274590580000026
to give the sequence D ═ (x)1,x2,x3…xn+1)
2): are used separately
Figure BDA0002274590580000027
To obtain the sequence B ═ (B)0,b1,b2,…bn)
3): by the sequence B ═ B0,b1,b2,…bn) And theorem 1.5 can yield:
Figure BDA0002274590580000028
Figure BDA0002274590580000029
Figure BDA00022745905800000210
Figure BDA0002274590580000031
4): finally, the plaintext M ═ a is solved0,a1,…an) Bob therefore deciphers the ciphertext that Alice sent to him.
Preferably, in the structure of the key system based on the quadratic residue provided by the invention, Bob finds out two large prime numbers p and q (which cannot be disclosed) in the encryption algorithm in the step (1), so that p ≡ q ≡ 3mod4 needs to be calculated, and simultaneously Blum number N ≡ p · q needs to be calculated, and Bob discloses N as the encryption key in the encryption algorithm.
Preferably, in the structure of the key system based on the quadratic residue provided by the invention, in the decryption algorithm in the step (2), Bob takes p and q as decryption keys in the decryption algorithm.
Has the advantages that:
compared with the prior art, the invention has the beneficial effects that: the invention aims to provide a structure of a key system based on secondary residue; has the advantages that: the cipher strength of the cipher system is not lower than that of RSA public key cipher system, and this can resist attack from selected plain text effectively and has high safety. The expansion rate of the ciphertext is not higher than that of the BG cryptosystem proposed by Blum and Goldwasser, and when the plaintext to be transmitted is longer, the expansion rate of the ciphertext is approximate to 1.
Drawings
FIG. 1 is a flow chart of an encryption algorithm
FIG. 2 is a flowchart of a decryption algorithm
Detailed Description
(1) Encryption algorithm
1): at ZnThe n-th irreducible polynomial p (x) a0+a1x+a2x2+…+anxnThe plaintext M is (a)0,a1,…an)，(a0,a1,…an) A sequence of coefficients that is an irreducible polynomial of degree n.
2): alice finds the encryption key N published in advance by Bob, and calculates as follows:
b0≡a0 2modN (1)
b1≡a1 2modN (2)
Figure BDA0002274590580000032
bn≡an 2modN (3)
to obtain the sequence B ═ (B)0,b1,b2,…bn)
3): a random variable k is introduced and the random variable k,
Figure BDA0002274590580000033
the following calculations were made:
x0≡k2modN (4)
x1≡x0 2modN (5)
Figure BDA0002274590580000041
xn+1≡xn 2modN (6)
to give the sequence D ═ (x)1,x2,x3…xn+1)
4): then calculate xn+2≡xn+1 2modN
5): calculating E ═ b0×x1,b1×x2,…bn×xn+1) Record c1＝b0×x1，c2＝b1×x2，…，cn+1＝bn×xn+1
5): transmitting ciphertext C ═ C1,c2,…cn+1,xn+2) To Bob.
(2) Decryption algorithm
1): bob obtains the ciphertext C ═ C (C) sent by Alice1,c2,…cn+1,xn+2) Then, according to theorem 1.5, we calculate in turn:
Figure BDA0002274590580000042
Figure BDA0002274590580000043
Figure BDA0002274590580000044
Figure BDA0002274590580000045
to give the sequence D ═ (x)1,x2,x3…xn+1)
2): are used separately
Figure BDA0002274590580000046
To obtain the sequence B ═ (B)0,b1,b2,…bn)
3): by the sequence B ═ B0,b1,b2,…bn) The following can be obtained:
Figure BDA0002274590580000047
Figure BDA0002274590580000048
Figure BDA00022745905800000410
4): finally, the plaintext M ═ a is solved0,a1,…an) Bob therefore deciphers the ciphertext that Alice sent to him.
The effect analysis of the invention:
(1) security analysis
It is easy to see from the encryption and decryption processes that if an enemy intercepts and captures the ciphertext and the encryption key and wants to obtain the related information of the plaintext, the square root under the modulus N must be calculated, the problem is solved as the problem of decomposing the product of the large sum of numbers as prime numbers, and therefore the security of the encryption algorithm is equal to that of the RSA encryption algorithm. Meanwhile, a random number k is added in the encryption process, so that the ciphertext has randomness. Namely, the same encryption key is used for the same plaintext, and the ciphertexts obtained when different random numbers are selected for encryption are different, so that the attack of plaintext selection is effectively prevented. Therefore, the security of the probability public key cryptosystem based on quadratic residue polynomial encryption is not lower than that of the RSA cryptosystem.
(2) Expansion ratio of ciphertext
The ratio of the ciphertext length to the plaintext length after encryption is defined as the expansion ratio of the ciphertext. In the cryptosystem constructed herein, the plaintext M is (a)0,a1,…an) The sequence is encrypted as C ═ C1,c2,…cn+1,xn+2) Sequence, ciphertext having a dilation Rate of
Figure BDA0002274590580000051
If Alice wants to transmit plaintext information particularly long, the expansion rate of the ciphertext can be seen as approximately 1. Generally, the cipher text expansion rate of the key system based on the quadratic residue structure is slightly larger than 1, but is lower than that of the BG cipher system proposed by Blum and Goldwasser.
(3) Encryption and decryption efficiency analysis
The time of the key system based on the secondary residual structure is mainly used for the computation of two square remainders, the decryption is used for the computation of two moduli, and the time complexity is slightly higher than that of a BG cryptosystem and is less than that of a public key cryptosystem O (k) such as RSA and the like3). Therefore, the encryption and decryption efficiency of the new probabilistic public key cryptosystem is high.
In summary, the construction method based on the secondary residual key system provided by the invention has higher security, and the ciphertext expansion rate reaches an ideal level.

Claims (3)

1. A structure based on a key system of quadratic residue is characterized in that: the method comprises the following steps:
(1) encryption algorithm
1): at ZnThe n-th irreducible polynomial p (x) a0+a1x+a2x2+…+anxnThe plaintext M is (a)0,a1,…an)，(a0,a1,…an) A sequence of coefficients that is an irreducible polynomial of degree n.
2): alice finds the encryption key N published in advance by Bob, and calculates as follows:
b0≡a0 2mod N (1)
b1≡a1 2mod N (2)
Figure FDA0002274590570000011
bn≡an 2mod N (3)
to obtain the sequence B ═ (B)0,b1,b2,…bn)
3): a random variable k is introduced and the random variable k,the following calculations were made:
x0≡k2mod N (4)
x1≡x0 2mod N (5)
Figure FDA0002274590570000013
xn+1≡xn 2mod N (6)
to give the sequence D ═ (x)1,x2,x3…xn+1)
4): then calculate xn+2≡xn+1 2mod N
5): calculating E ═ b0×x1,b1×x2,…bn×xn+1) Record c1＝b0×x1，c2＝b1×x2，…，cn+1＝bn×xn+1
5): transmitting ciphertext C ═ C1,c2,…cn+1,xn+2) To Bob.
(2) Decryption algorithm
1): bob obtains the ciphertext C ═ C (C) sent by Alice1,c2,…cn+1,xn+2) Then, according to theorem 1.5, we calculate in turn:
Figure FDA0002274590570000014
Figure FDA0002274590570000015
Figure FDA0002274590570000016
Figure FDA0002274590570000017
to give the sequence D ═ (x)1,x2,x3…xn+1)
2): are used separately
Figure FDA0002274590570000021
To obtain the sequence B ═ (B)0,b1,b2,…bn)
3): by the sequence B ═ B0,b1,b2,…bn) And theorem 1.5 can yield:
Figure FDA0002274590570000022
Figure FDA0002274590570000023
Figure FDA0002274590570000024
Figure FDA0002274590570000025
4): finally, the plaintext M ═ a is solved0,a1,…an) Bob therefore deciphers the ciphertext that Alice sent to him.
2. The structure of a secondary residual-based key system according to claim 1, wherein: in the encryption algorithm of step (1), Bob finds out two large prime numbers p and q (which cannot be disclosed), so that p ≡ q ≡ 3mod4 needs to be calculated, and simultaneously, Blum number N ≡ p · q needs to be calculated, and Bob discloses N as an encryption key in the encryption algorithm.
3. The structure of a secondary residual-based key system according to claim 1, wherein: in the decryption algorithm in the step (2), Bob takes p and q as decryption keys in the decryption algorithm.
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Citations (3)

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Publication number Priority date Publication date Assignee Title
US8385541B2 (en) * 2010-02-18 2013-02-26 King Fahd University Of Petroleum And Minerals Method of performing elliptic polynomial cryptography with elliptic polynomial hopping
CN104184581A (en) * 2014-08-07 2014-12-03 中国人民武装警察部队工程大学 Method for encrypting public key in signed quadratic residue class group
CN106341230A (en) * 2015-07-08 2017-01-18 吴清山 Unconditional security cryptosystem

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US8385541B2 (en) * 2010-02-18 2013-02-26 King Fahd University Of Petroleum And Minerals Method of performing elliptic polynomial cryptography with elliptic polynomial hopping
CN104184581A (en) * 2014-08-07 2014-12-03 中国人民武装警察部队工程大学 Method for encrypting public key in signed quadratic residue class group
CN106341230A (en) * 2015-07-08 2017-01-18 吴清山 Unconditional security cryptosystem

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
刘直良: "一种基于二次剩余的改进的RFID认证协议", 《电信科学》 *
孙国栋等: "基于REESSE1+公钥密码体制的概率加密", 《计算机工程》 *
曹军委等: "基于Java的RSA公钥密码算法的实现与探讨", 《软件导刊》 *
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Application publication date: 20200228