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DES are now practical During 1997, it would take a few months to break DES using the brute force approach, while in 1999, an attempt to do so succeeded in 22 hours In addition several theoretical attacks that can break DES have also been demonstrated Thus, a replacement for DES was needed This replacement is the AES algorithm, which was selected in October 2000 AES has been designed to be resistant to known attacks and exhibits simplicity of design Originally called as Rijndael, this was issued as a Federal Information Processing Standards (FIPS) standard in November 2001 AES uses three different key sizes, namely 128, 192, and 256 bits The block size is 128 bits It treats data in four groups of four bytes, each called the state AES has been designed to have one of 9, 11, or 13 rounds In each round the state undergoes four operations, namely byte substitution, shifting of rows, mixing of columns and XORing with the subkey of the round All operations can be combined into XOR and table lookups Hence these are very fast and ef cient DES and AES are not the only block ciphers widely considered In fact, there are several others, such as Triple-DES, Blow sh, IDEA etc No block cipher is ideally suited for all applications There are several factors that go into the selection of a cryptographic algorithm for a given scenario These include the desirable key size, block size, complexity of cryptographic mapping, tolerance to error, and the estimated security level As explained earlier, longer key sizes are desirable but this imposes additional costs associated with generation, transmission, and storage An interesting technique used by block ciphers such as Khufu and stream ciphers such as SEAL is to make the task of changing cipher keys computationally expensive while allowing encryption to remain relatively ef cient Larger block size is desirable but might be more costly to implement It might also affect performance, say by requiring padding Complexity of cryptographic mapping affects implementation and development costs as well as the realtime performance In fact some ciphers are more ef cient when implemented in hardware and other ciphers are more ef cient when implemented in software The effects of corrupt ciphertext during decryption indicate the error propagation characteristics of the cipher This is also an important factor to consider given the lossy wireless links in an ad hoc environment The estimated security level of a cipher can be assumed to be proportional to the amount of expert cryptanalysis withstood 2223 Asymmetric Encryption Asymmetric encryption represents a very signi cant advance in the three thousand year history of cryptography We have seen earlier that conventional symmetric key cryptography uses one key that is shared by both the sender and the receiver Disclosure of this key will result in compromise of communications In addition, this approach affords no protection if a receiver forges a message and claims such a message is from the sender This is possible because two parties share the key and either of them could create a message using the shared key Thus, symmetric key schemes do not lend themselves easily to verifying the authenticity of the sender Another problem associated with such schemes is related to the secure distribution of keys These problems can be addressed using the approach of asymmetric key cryptography The asymmetric approach uses two keys, a public key and a private key The public key may be known to anybody while the private key is supposed to be known only to the entity creating a message Every entity in the network wishing to send messages has these two keys, namely a private and a public key Both the keys are different but related mathematically Additionally, it is computationally infeasible to derive the private key knowing only the public key and the cryptographic algorithm Derivation of the private.

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(N ) (N ) (N ) the value of h at the jump moments by assuming H0 = fN ( 1 10A, a1 5A) a (N ) (N ) (N ) (N ) and H = fN ( + A/3, a +1 A/3) for = 1, , N 1 and H N = a (N ) (N ) (N ) (N ) fN ( 1 +5A, a1 +10A) To nish, we build the estimator N = ( 1 , , a N ; a a (N ) (N ) H0 , , H N ), whose consistency is established in [BEN 00]

THEOREM 611 Qh is a step fractional Brownian process of function of scale h( ) verifying (652) If, moreover, A < 0 and ] 0 /2, 0 [, then we have limN + N = 0 as with 0 = (a1 , , aK ; H0 , , HK ) 66 Bibliography

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