m = c d mod n：m = c d mod n

Cryptography ,the block size is i bits, where 2i n= 2i+1C = Me mod nM = Cd mod n = (Me)d mod n = Med mod npublic key of KU = e, nprivate key of KR = d, nRequirementsIt is possible to find values of e, d, n such that Med = M mod n for all M n.It is relatively easy to calculate mod Me and Cd for all values of M n.It is infeasible to determine d given e and n.Med = M mod nAccording to Eulers theoremGiven two prime numbers p 1 e (n) (public, chosen)d =e-1(mod (n) (private, calculated)The private key consists of d, n and the public key consists of e, n.Suppose that user A has published its public key and that user B wishes to send the message M to A. Then B calculates C = Me mod n and transmits C. On receipt of this ciphertext, user A decrypts by calculating M = Cd mod n. Select two prime numbers, p = 17 and q = 11. Calculate n = pq = 17 x 11 = 187. Calculate (n) = (p - 1)(q - 1) = 16 x 10 = 160. Select e such that e is relatively prime to (n) = 160 and less than (n) we choose e = 7. Determine d such that de = 1 (mod 160) and d 160. The correct value is d = 23, because 23 x 7 = 161 = 160 + 1; d can be calculated using the extended Euclids algorithmRSA is usch slower than DES otherwise, go to step 2.Issues in RSARSA Factoring Challenge - practical difficulty of factoring large integers and cracking RSA keys used in cryptographyThe smallest , a 100 decimal digit number called RSA-100 was factored by April 1, 1991A primary application is for choosing the key length of the RSA public-key encryption scheme. The RSA numbers were generated on a computer with no network connection of any kind. The computers hard drive was subsequently destroyed so that no record would exist, anywhere, of the solution to the factoring challenge.The first RSA numbers generated, from RSA-100 to RSA-500, were labeled according to their number of decimal digitsDH Computation IssuesDiffieHellman key exchange (DH) is a cryptographic protocol that allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher.Alice and Bob agree to use a prime number p=23 and base g=5. Alice chooses a secret integer a=6, then sends Bob A = ga mod p A = 56 mod 23 = 8. Bob chooses a secret integer b=15, then sends Alice B = gb mod p B = 515 mod 23 = 19. Alice computes s = B a mod p 196 mod 23 = 2. Bob computes s = A b mod p 815 mod 23 = 2. Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.Elliptic curves are also used in several integer factorization algorithms that have applications in cryptographyAn elliptic curve is a plane curve which consists of the points satisfying the equationy2 = x3 + ax + bSeveral RSA-based protocols have been adapted to elliptic curves, replacing the group Zpq with an elliptic curve:* The Elliptic Curve Diffie-Hellman key agreement scheme is based on the Diffie-Hellman scheme,* The Elliptic Curve Digital Signature Algorithm is based on the Digital Signature Algorithm,* The ECMQV key agreement scheme is based on the MQV key agreement scheme.HASH Function in CryptographyA hash function H is a transformation that takes a variable-size input m and returns a fixed-size string, which is called the hash value h (that is, h = H(m). Hash functions with just this property have a variety of general computational uses, but when employed in cryptography the hash functions are usually chosen to have some additional properties. The basic requirements for a cryptographic hash function are:* the input can be of any length,* the output has a fixed length,* H(x) is relatively easy to compute for any given x ,* H(x) is one-way,* H(x) is collision-free.Used in digital signatures, message authentication codes (MACs). They can also be used as ordinary hash functions, to index data in hash tables; as fingerprints, to detect duplicate data or uniquely identify filesA related application is password verification. Passwords are usually not stored in cleartext, instead in digest form. To authenticate a user, the password presented by the user is hashed and compared with the stored hash. This is sometimes referred to as one-way encryption.MD5 algorithmIn cryptography, MD5 (Message-Digest algorithm 5) is a widely used cryptographic hash function with a 128-bit hash value. commonly used to check the integrity of files. MD5 processes a variable-length message into a fixed-length output of 128 bits. The input message is broken up into chunks of 512-bit blocks (sixteen 32-bit little endian integers)Chinese Remainder TheoremThere are certain things whose number is unknown.Repeatedly divided by 3, the remainder is 2;by 5 the remainder is 3; and by 7 the remainder is 2.What will be the number? Suppose n1, n2, , nk are positive integers which are pairwise coprime. Then, for any given in