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THE PUBLIC KEY TREASURE CHEST
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We provide a treasure chest analogy to explain how asymmetric cryptography works. If Bob wants people to be able to send him messages secretly, he can ask them to put these messages inside of a treasure chest, and give him the treasure chest. Then he can open the treasure chest. Imagine that Bob can go to some public area such as a park and leave as many open treasure chests as he wants. When people want to send Bob a message, they can put the message into the chest and close it. Anyone is able to lock the treasure chest just by closing it. Bob can come to the park at some point and look at which treasure chests have been locked. Bob has a key that he can use to unlock the treasure chests to retrieve the private messages that people left for him. This basic idea is summarized in the following illustration. In essence, in asymmetric cryptography, when Bob gives out his public key, it is like him giving out an open, empty treasure chest. Anybody can put a message in the treasure chest and lock it by encrypting with Bob s public key. Bob is the only one who can open the treasure chest because his private key is the key to the treasure chest. When Bob does a decryption with his private key, it corresponds to unlocking the chest.
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CHAPTER 13 s ASYMMETRIC KEY CRYPTOGRAPHY
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To extend the notation first described in Section 12.1.2 for asymmetric cryptography, we introduce two keys: kp and ks, which are the public and private (or secret) keys, respectively. An encryption function takes a message and the public key as input to produce ciphertext: F(m,kp) = c. The decryption function takes the ciphertext and private key as input, and produces the plaintext: F 1(c,ks) = m. An asymmetric cipher, then, is an encryption and decryption function for which F 1(F(m,kp),ks) = m. The cipher is asymmetric because different keys are used for encryption and decryption.
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13.2. RSA
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After briefly discussing how symmetric ciphers worked in 12, we provided some examples of them, such as DES and AES. Now that you have learned how asymmetric ciphers work in general, we give two examples of them; namely, RSA and ECC. RSA was the first asymmetric encryption algorithm ever published. Shortly after Diffie and Hellman published a paper about the idea of an asymmetric cipher, Rivest, Shamir, and Adelman (the R, S, and A, in RSA) came up with a concrete algorithm that was able to serve as an asymmetric encryption scheme. RSA is the most widely known and used asymmetric cipher. It is used in a variety of different protocols in the world of computer security, including SSL, CDPD,1 and PGP 2 RSA has . been used in many different applications to date, and is likely to be used in many different applications in the future. The mathematical properties of the RSA algorithm are based on number theory. The security of the algorithm depends on the difficulty of factoring large prime numbers. If it is difficult to factor large prime factors, it will be hard to break the mathematical properties of the algorithm. Common key sizes that are used with RSA are 1024, 2048, and 4096 bits. Since RSA is fundamentally different than any other encryption algorithm, these key sizes do not have a direct relation to the key sizes of other algorithms, such as AES or Triple DES. That is, a message encrypted with a 2048-bit RSA key may not be any more or less secure than a message encrypted with a 256-bit AES key. You could attempt to compare the strength of key sizes of different algorithms by measuring the expected amount of time it would take to successfully conduct a brute-force attack on them. (Read A.K. Lenstra s paper, Selecting Cryptographic Key Sizes for guidance on key sizes.) Yet, in general, it may not make much sense to directly compare the lengths of keys of two different algorithms.
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13.3. Elliptic Curve Cryptography (ECC)
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Elliptic curve cryptography (ECC) provides another mathematical way to build a public key cryptosystem. It was invented to by Neil Koblitz and Victor Miller independently at about the same time in 1985. Its discovery is much more recent than RSA. ECC is also based on number theory. Unlike RSA, its security is not dependent upon the difficulty of factoring large prime numbers, but instead is based upon the difficulty of the elliptic curve discrete logarithm problem.
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1. Cellular Digital Packet Data (CDPD) is a wireless data protocol. 2. PGP (Pretty Good Privacy) is a secure e-mail system.
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