of factorization of large numbers (or finding logarithms) mean in terms of resource, time, and effort. The fact that teams were able to complete the factorization of RSA-140 and RSA-155/512 does not mean that the RSA cryptosystem is now broken or "cracked." Instead, these successful experiments provided valuable information on the state of the art in factorization, from algorithms and methods to hardware and time requirements. It is still safe for Alice and Bob to use 512-bit RSA keys (even more SO with 1024-bit keys), knowing the costs in personnel, capital investment, and time that Eve would have to support. Another important consideration is the lifetime of the data to be protected. This lifetime can range from the scale of a single day (e.g., certified signature, restricted-access broadcast, stock-exchange instructions) to several years (e.g., business and financial contracts, long-term strategy). With very large scale integration (VSLI), integrated components (IC) technology has produced chips capable of performing RSA encryption and decryption with various modulus sizes (namely 32, 120, 256, 272, 298, 512, 593, 1024). Compared with DES (see the next section), RSA is 1000 times slower from the hardware perspective. From the software perspective, RSA is only 100 times slower. Encryption is always faster than decryption. Using a workstation, the encryption and decryption tasks require 30 ms and 160 ms, respectively, for 512 bits and 80 ms and 930 ms for 1024 bits. 11 The process can be speeded up by an appropriate choice of public key. Appropriate values are those which have binary words of value 2N + 1, which in binary have the form (100...01) with N + 1 bits and only two 1 bits. This choice reduces the operation of exponentiation to only N + 1 successive multiplications. Referring back to the PKC parameters introduced in Section 20.7, the standard recommendations are e = 3(11), e = 17(10001), and e = 65537(100001). Such a selection avoids e = 5(101) and e = 9(1001) because of the fact that the public key mus