3This section uses some elementary concepts from number theory. Far a review, see Appendix B. in .NET Include Code 39 in .NET 3This section uses some elementary concepts from number theory. Far a review, see Appendix B.

How to generate, print barcode using .NET, Java sdk library control with example project source code free download:

3This section uses some elementary concepts from number theory. Far a review, see Appendix B. generate, create code-39 none for .net projects Code 128 Code Set A 3.4 / PUBLIC-KEY CRYPTOGRAPHY ALGORITHMS Figure 3.8 The RSA Algorithm divisorof e and <J>(n) 1]. Finally, calculate d as the multiplicative inverse of e, is modulo <J>(n). can be shown that d and e have the desired properties. l Suppose that user A has published its public key and that user B wishes to sendthe message M to A.

Then B calculates C = Me (mod n) and transmits C. On. receiptof this ciphertext, user A decrypts by calculating M cI (modn).. An example, from [SING99 Visual Studio .NET barcode 3 of 9 ], is shown in Figure 3.9.

For this example, the keys were generated as follows:. 1. Select two prime numbers, p = 17 and q = 11. 2. Calculate n = pq = 17 X 11 = 187. 3. Calculate <J>(n) (p - l)(q - 1) = 16 = e X 10. = 160.. 4. Select e such that e is relatively prime to <J>(n) 160 and less than <J>(n); choose = we = 7.. S. Determine d such that de mod 160 = 1 and d < 160. The correct value is d = 23,.

because 23 X 7 = 161 = 10 X 160 + 1. CHAPTER 3 I PUBLIC-KEY CRYPTOGRAPHY AND MESSAGE AUTHENTICATION The resulting keys are p ublic key KU = {7,187}and private key KR = {23,187}. The example shows the use of these keys for a plaintext input of M = 88. For encryption, we need to calculate C = 887 mod 187.

Exploiting the properties of modular arithmetic, we can do this is folIows:. 887 mod 881 mod 882 mod .NET barcode code39 884 mod 887 mod 187 = [(884mod 187) X (882 mod 187) X (881mod 187)] mod 187 187 = 88 187 = 7744 mod 187 = 77 187 = 59,969,536 mod 187 = 132 187 = (88 X 77 X 132) mod 187 = 894,432mod 187 = 11. For decryption, we calcu late M = 1123mod 187: 1123mod 187 = [(111mod 187) X (112 mod 187) X (114 mod 187) X (118 mod 187) X (118 mod 187)] mod 187 111mod 187 = 11 112mod 187 = 121 114mod 187 = 14,641 mod 187 = 55 118mod 187 = 214,358,881 mod 187 = 33 1123mod 187 = (11 X 121 X 55 X 33 X 33) mod 187 = 79,720,245 mod 187 = 88 There are two possible approaches to defeating the RSA algorithm. The first is the brute-force approach: Try alI possible private keys. Thus, the larger the num-~ ber of bits in e and d, the more secure the algorithm.

However, because the calculations involved, both in key generation and in encryption/decryption, are complex, the larger the size of the key, the slower the system wilI run. Most discussions of the cryptanalysis of RSA have focused on the task of factoring n into its two prime factors. For a large n with large prime factors, factoring is a hard problem, but not as hard as it used to be.

A striking illustration of this occurred in 1977; the three inventors of RSA challenged Scientific American ers to decode a cipher they printed in Martin Gardner"s "Mathematical Games" umn [GARD77]. They offered a $100 reward for the return of a plaintext an event they predicted might not occur for some 40 quadrillion years. In ApriI 1994, a group working over the Internet and using over 1600 computers claimed prize after only eight months of work [LEUT94J; This challenge used a public-key~.

Encryption Decryption KU = 7, 187 Figure 3.9 Example af RSA Algarithm KR = 23, 187. 3.4 / PUBLlC-KEY CRYPTOGRAPHY ALGORITHMS size (length of n) of 12 9 decimaI digits, or around 428 bits. This resuIt does not invaIidate the use of RSA; it simply means that larger key sizes must be used. Currently, a 1024-bit key size (about 300 decimaI digits) is considered strong enough for virtually all applications.

Diffie-Hellman Key Exchange. The first pubIished publ visual .net Code 3 of 9 ic-key aIgorithm appeared in the seminaI paper by Diffie and Hellman that defined public-key cryptography [DIFF76] and is generally referred to as Diffie-Hellman key exchange. A number of commerciaI products employ this key exchange technique.

The purpose of the aIgorithm is to enabIe two users to exchange a secret key securely that can then be used for subsequent encryption of messages. The algorithm itseIf is limited to the exchange of the keys. The Diffie-Hellman algorithm depends for its effectiveness on the difficulty of computing discrete logarithms.

Briefly, we can define the discrete logarithm in the following way. First, we define a primitive root of a prime number p as one whose powers generate all the integers from 1 to p-L That is, if a is a primitive root of the prime number p, then the numbers a modp, a2 modp,..

., aP-1 modp. are distinct and consist of the integers from 1 through p-l in some permutation. For any integer b less than p and a primitive root a of prime number p, one can find a unique exponent i such that b = ai mod p where O ::s i ::s (p - 1). The exponent i is referr Code 39 for .NET ed to as the discrete logarithm, or index, of b for the base a, modp. This value is denoted as inda,p(b).

With this background we can define the Diffie-Hellman key exchange, which is summarized in Figure 3,10. For this scheme, there are two publicly known numbers: a prime number q and an integer a that is a primitive root of q. Suppose the users A and B wish to exchange a key.

User A selects a random integer XA < q and computes YA = aXA mod q. Similarly, user B independently selects a random integer XB < q and computes YB = aXB mod q. Each side keeps the X value private and makes the Y value available publicly to the other side.

User A computes the key as K = (yB)XA mod q and user B computes the key as K = (yA)XB mod q. These two calculations produce identical results: K = = = = = = = (yB)XA mod q (aXB mod q)XA mod q (aXB)XAmod q aXBXAmod q (aXA)XBmod q (aXA mod q)XB mod q (yA)XB mod q.
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