Book: introduction of cryptography and security CRYPTOGRAPhY 7.14. The modulus o

Question

Book: introduction of cryptography and security CRYPTOGRAPhY 7.14. The modulus of RSA has been enlarged over the years in order to thwart im- proved attacks. As one would assume, public-key algorithms become slower as the modulus length increases. We study the relation between modulus length and perfor- mance in this problem. The performance of RSA, and of almost any other public-key algorithm, is dependent on how fast modulo exponentiation with large numbers can be performed. 1. Assume that one modulo multiplication or squaring with k-bit numbers takes c k clock cycles, where c is a constant. How much slower is RSA encryp- tion/decryption with 1024 bits compared to RSA with 512 bits on average? Only consider the encryption/decryption itself with an exponent of full length and the square-and-multiply algorithm. 2. In practice, the Karatsuba algorithm, which has an asymptotical complexity that is proportional to kog23, is often used for long number multiplication in cryptog- raphy. Assume that this more advanced technique requires k3 -k585 clock cycles for multiplication or squaring where ' is a constant. What is the Problems 203 ratio between RSA encryption with 1024 bit and RSA with 512 bit if the Karat- suba algorithm is used in both cases? Again, assume that full-length exponents are being used.

Explanation / Answer

RSA Algorithm in Coding RSA algorithm is asymmetric cryptanalysis algorithm. Asymmetric authentically denotes that it whole kit and boodle on two different paint stone s i.e. Public Key and Private Key. As the identification describes that the Public Key is given to everyone and Private winder stone is kept private . An example of asymmetric cryptography : A customer (for example web browser ) sends its public key to the server and requests for some information . The server encrypts the data utilizing client’s public key and sends the encrypted data. Client receives this data and decrypts it. Since this is asymmetric, nobody else except browser can decrypt the data even if a third company has public key of browser. The construct ! The conception of RSA is predicated on the fact that it is arduous to factorize a sizably voluminous integer. The public key consists of two phone number s where one number is multiplication of two immensely colossal prime routine game . And private key is additionally derived from the same two prime numbers. So if somebody can factorize the immensely colossal number, the private key is compromised. Consequently encryption vigour thoroughly lies on the key size and if we double or triple the key size, the zip of encryption increases exponentially. RSA paint can be typically 1024 or 2048 bite long, but expert believe that 1024 bit keys could be jailbreak in the near future. But till now it seems to be an infeasible task.

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