Research On Nonlinear Theory In Secure Communication

Secure communication technology is involved in the transmission of information safe and efficient, and concerned with national security and the national economy and the people's livelihood. At the same time, nonlinear science is concerned about exploring the law of complex system, and become the focus and hotspot of researching. Therefore, it is significant theoretically and practically to adapt non-linear theory to constructing the system of secure communication.Since 1980s, the idea of using chaotic systems to design secure communication system has attracted more and more attention. The use of chaos in cryptography depends on the natural relation between the two research areas: strong dynamical properties of chaotic systems imply strong cryptographical properties of cryptosystems, and the basic way to make cryptosystems with good strong cryptographical properties implies quasi-chaos. Generally speaking, there are two paradigms of applying chaos for secure applications: symmetrical secure communication and asymmetrical secure communication. This dissertation firstly focuses on asymmetrical secure communication, i.e., the area lying between chaos theory and pure cryptography in asymmetrical secure communication system. Meanwhile, we apply non-linear theory to information hiding and digital watermark and gain some elementary results.Our contributions in this dissertation involve the following three aspects about non-linear secure communication: theoretical analyses results in a one-way trapdoor function based on extended Chebyshev polynomials and the verification of the feasibility and security. Further few secure algorithms are presented. In the end, the methods of information hiding and fragile digital watermark are researched, and the algorithms about them based on multi-chaos map present. The main achievements contained in this dissertation are as follows:1. In 2003, Ljupco Kocarev etc used the properties of chaos and semi-group, and proposed a public-key cipher based on Chebyshev polynomials. But any attacker easily gets the plaintext message given known ciphertext for the inherent periodic property of trigonometric function. Thus, an improved public-key encryption project which is based on Chebyshev polynomials, is presented.2. It is investigated for the properties of Chebyshev polynomials and extended Chebyshev iteration formula, and the difficulty of discrete Chebyshev problem. And the conclusion has been reached that if the iteration parameters are integer, the compute complexity of extended Chebyshev polynomials is the same as discrete logarithm problem and integer factoring problem.Because that the public-key encryption scheme based on extended Chebyshev polynomials doesn't need a large prime, the efficiency is higher than conventional RSA and EIGamal algorithms, and at least no lower than them.3. By combining Chebyshev polynomials with modulus compute, extending Chebyshev polynomials' definition domain to real number, the following conclusion has been drawn by theoretic verification and data analyses: based on the properties of extended Chebyshev polynomials in real number, the proposed break scheme from Pina Bergamo et al can be avoided. And it is feasible to use extended Chebyshev polynomials to construct public-key encryption in theory.4. A one-way trapdoor function based on the properties of semi-group and modulus compute of Chebyshev polynomials iteration formula has been constructed and applied to public-key cryptography. The relative public-key encryption, key exchange, authentication identity and digital signal schemes are presented. These are proven as feasible and efficient.5. In 2005, Di Xiao proposed an entire deniable protocol based on chaotic maps, which used chaotic model to a chaotic key exchange scheme and a chaotic hash function. After the analysis of the algorithms, the scheme above is insecure. Thus an improved scheme is presented.6. A single Logistic map is analysed, the fault of its application to the field of information hiding and digital watermarking found, and the relative optimal project based on multi-chaos map present.To use the property of dependence on initial value,we present the digital hiding algorithm based on multi-chaos map and image mergence.It is proven that the algorithm is robust,secure.7. An fragile watermarking with high precision based on multi-chaotic model is proposed, and detection accuracy is also proposed and analyzed. The fragile watermark algorithm, which uses chaotic systems to produce watermark dynamically, can control location precision and resist pse- udo-authentication. Tampering estimation algorithm in spatial and frequency domain based on integer DWT domain not only can decide the position of spatial tampering, but also estimate the sorts of filtering that the image has undergone. The integrity authentication process is completely blind which requires neither the original image nor the original watermark.