Different from traditional symmetric cryptography, public key cryptography allows each user to own a pair of public/private key pair. Given the public key of one user and the data to be encrypted, a ciphertext can be ...Different from traditional symmetric cryptography, public key cryptography allows each user to own a pair of public/private key pair. Given the public key of one user and the data to be encrypted, a ciphertext can be generated such that only the intended user is able to recover the data using his/her private key associated with public key involved in the encryption process. To really understand the philosophy behind the public key cryptography is not an easy task and sometimes it seems somewhat difficult, if not impossible, for the beginners without the knowledge of the abstract algebra and number theory. In this paper, we introduce briefly the basic definition of public key cryptography as well as the corresponding mathematical hard problems. Besides, we explain why the mathematical backgrounds, especially abstract algebra and number theory, matter in the study of public key cryptography.展开更多
In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physica...In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physical entities and more associated with mental events, thinking shifts from analytical to conceptual-abstract. Fundamental topics taken from the abstract algebra (aka: modern algebra) are unquestionably abstract. Historically, fundamental concepts taught from the abstract algebra are detached from physical reality with one exception: Boolean operations. Even so, many abstract algebra texts present Boolean operations from a purely mathematical operator perspective that is detached from physical entities. Some texts on the abstract algebra introduce logic gate circuits, but treat them as perceptual symbols. For majors of pure or applied mathematics, detachments from physical entities is not relevant. For students of Computer and Electrical Engineering (CpE/EE), mental associations of Boolean operations are essential, and one might argue that studying pure Boolean axioms are unnecessary mental abstractions. But by its nature, the CpE/EE field tends to be more mentally abstract than the other engineering disciplines. The depth of the mathematical abstractions that we teach to upper-division CpE/EE majors is certainly up for questioning.展开更多
文摘Different from traditional symmetric cryptography, public key cryptography allows each user to own a pair of public/private key pair. Given the public key of one user and the data to be encrypted, a ciphertext can be generated such that only the intended user is able to recover the data using his/her private key associated with public key involved in the encryption process. To really understand the philosophy behind the public key cryptography is not an easy task and sometimes it seems somewhat difficult, if not impossible, for the beginners without the knowledge of the abstract algebra and number theory. In this paper, we introduce briefly the basic definition of public key cryptography as well as the corresponding mathematical hard problems. Besides, we explain why the mathematical backgrounds, especially abstract algebra and number theory, matter in the study of public key cryptography.
文摘In mathematics, physics, and engineering, abstract concepts are an indispensable foundation for the study and comprehension of concrete models. As concepts within these fields become increasingly detached from physical entities and more associated with mental events, thinking shifts from analytical to conceptual-abstract. Fundamental topics taken from the abstract algebra (aka: modern algebra) are unquestionably abstract. Historically, fundamental concepts taught from the abstract algebra are detached from physical reality with one exception: Boolean operations. Even so, many abstract algebra texts present Boolean operations from a purely mathematical operator perspective that is detached from physical entities. Some texts on the abstract algebra introduce logic gate circuits, but treat them as perceptual symbols. For majors of pure or applied mathematics, detachments from physical entities is not relevant. For students of Computer and Electrical Engineering (CpE/EE), mental associations of Boolean operations are essential, and one might argue that studying pure Boolean axioms are unnecessary mental abstractions. But by its nature, the CpE/EE field tends to be more mentally abstract than the other engineering disciplines. The depth of the mathematical abstractions that we teach to upper-division CpE/EE majors is certainly up for questioning.
