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bears his signature. Along with NSAM 160 is the Weisner Memorandum, which includes more details about permissive links. It can be inferred that the authors proposed equipping nuclear arms with cryptographic switches. Bombs could be activated only with the correct codes, with a form of public-key crypto guaranteeing correct codes (two principles referred to as authentication and nonrepudiation; see 5). What about the former Soviet Union or the People s Republic of China Did these nations have public-key algorithms before 1976 Or how about Hungary or Japan or any other government If they did, they re not saying.
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Figure 4-7 In symmetric-key crypto, generally the last thing done in encrypting is the first thing done (in reverse) in decrypting
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and manipulates them using computer operations, public-key crypto operates on the data as numbers and plays with the numbers (see Figure 4-8). And the math is one-way: It s easy in one direction but not in the other direction. In fact, the foundation of any good public-key algorithm is a oneway function, the class of math problems on which public-key crypto is built. Actually, public-key one-way functions are more accurately described as one-way with a trap door. To the rest of the world the functions are one-way, but the private key operates as a trap door that allows the owner to recover the original data (see Figure 4-9). There are true oneway functions, and we talk about some of them in 5.
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The Key Distribution Problem and Public-Key Cryptography
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Figure 4-8 Public-key crypto treats all data as numbers and performs mathematical operations on them
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A number 338,638,982
ASCII 0x47 0x6F 0x61 0x6C
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In this book, we don t describe the full details of the math behind the various algorithms; you can find that in the RSA Labs FAQ on the accompanying CD. But in the following sections we talk about the three most widely used algorithms for solving the key distribution problem: RSA, DH, and ECDH (Elliptic Curve Diffie-Hellman). We tell you the names of the one-way functions and outline the problems.
Figure 4-9
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A one-way function with a trap door. Performing operations in one direction is easy, but reversing the steps is difficult unless you know the secret trap door
The RSA Algorithm
The RSA algorithm encrypts data. If you feed your plaintext to the algorithm along with the public key, you get ciphertext as a result. With the digital envelope, the plaintext is the session key. It s certainly possible to use RSA to encrypt data other than a session key, but RSA is not as fast as the symmetric algorithms. For example, RC4 (probably the fastest symmetric algorithm in wide use today) will encrypt data at a rate 700 times faster than 1,024-bit RSA (1,024 bits is the most commonly used RSA key size). RC5 (one of the fastest block ciphers) is about 500 times faster.
NOTE:
Incidentally, the R in RC4 and RC5 is the same R as in RSA. So the best way to use RSA is to create a digital envelope. For example, Pao-Chi can generate a random or pseudo-random 128-bit RC4 key, use it to encrypt his e-mail message to Gwen, and then use Gwen s RSA public key to encrypt the RC4 key. Encrypting the RC4 key (16 bytes) will take only a few milliseconds on most platforms. Pao-Chi s message to Gwen consists of two parts: the encrypted session key and the encrypted bulk data (see Figure 4-10). Gwen separates the two components, uses her RSA private key to decrypt the session key, and then uses that decrypted RC4 key to decrypt the bulk data.
AM FL Y
The Key Distribution Problem and Public-Key Cryptography
Figure 4-10 In Pao-Chi s message, the encrypted session key comes first and the encrypted bulk data follows
An RSA public key consists of two numbers: a modulus and a public exponent. The private key is made up of the same modulus and a private exponent (see Figure 4-11). The modulus, incidentally, is the product of two very large prime numbers. (A prime number, or prime, cannot be evenly divided; for example, 3, 5, 7, 13, and 17 are primes.) In the cryptographic literature, these numbers are usually given the romantic names n, e, and d, where n is the modulus, e is the public exponent, and d is the private exponent. Equally poetic are the names for the two primes that make up the modulus: p and q. When you generate an RSA key pair (or rather, when the program you re running generates an RSA key pair), you decide on a public exponent e, find two large primes p and q that work with the e you ve chosen, multiply p and q to get the modulus n, and finally compute your private
Figure 4-11
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