It is well known that the degree of a 2m-variable bent function is at most m. However, the case in homogeneous bent functions is not clear. In this paper, it is proved that there is no homogeneous bent functions of degree m in 2m variables when m > 3; there is no homogenous bent function of degree m − 1 in 2m variables when m > 4; Generally, for any nonnegative integer k, there exists a positive integer N such that when m > N , there is no homogeneous bent functions of degree m − k in 2m variables. In other words, we get a tighter upper bound on the degree of homogeneous bent functions. A conjecture is proposed that for any positive integer k > 1, there exists a positive integer N such that when m > N , there exists homogeneous bent function of degree k in 2m variables.
In order to distinguish between human users and computer programs, CAPTCHA (Completely Automated Public Turing test to tell Computers and Human Apart) mechanism is widely applied in websites such as accounts application website. While the major implementation of CAPTCHA method-2D still image verification code based on OCR technology is threatened by developing artificial intelligence and image recognition technologies. In this paper, we propose a new approach to implement CAPTCHA mechanism based on 3D Animation, utilizing the weakness of computer vision, which make it robust to computer attacks and convenient for users to recognize, and implemented this method to generate a 3D animation verification code.
Nowadays, malicious computer programs have attempted to attack websites, greatly affecting their availability and security. CAPTCHA (Completely Automated Public Turing test to tell Computers and Human Apart) mechanism is an effectual method in solving this problem. The prevailing implementation of CAPTCHA is 2D still image verification code, however, the developing artificial intelligence and image recognition technologies make it possible for computer programs to pass through CAPTCHA's test. In this paper, we propose a new CAPTCHA implementation, in the form of 3D animation and based on the weak point of computer vision. This new method could prevent attacks based on both image recognition and moving objects recognition in videos. Coupled with easy use for humans, it proves to be a kind of novel, safe and practical CAPTCHA.
By the relationship between the Walsh spectra at partial points and the Walsh spectra of its sub-functions, by the action of general linear group on the set of Boolean functions, and by the Reed-Muller transform, a novel method is developed, which can theoretically construct all bent functions. With this method, we enumerate all bent functions in 6 variables; in 8variable case, our method is more efficient than the method presented by Clark though we still can not enumerate all bent functions; enumeration of all homogeneous bent functions of degree 3 in eight variables can be done in one minute by a P4 1.7G HZ computer; construction of homogenous bent function of degree 3 in 10 variables is efficient too; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is proved.
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