Public key crypto-systems in .NET Connect code-128c in .NET Public key crypto-systems

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11.6 Public key crypto-systems use none none integration toconnect none with none Specific Terms for GS1 Barcodes The private k none for none ey crypto-systems we have considered previously have relied on the distribution of key material. There is another approach that comes under the heading of public key crypto-systems, which does not require the prior secure communication of key material. Perhaps the most widely used one is the RSA (for Rivest, Shamir and Adleman [14]) system.

The RSA crypto-system is based on the fact that in order to factorize a number with N digits, a classical computer needs to perform a number of steps that at least grows faster than polynomially with N. Consider the factorizing of the numbers 21 and 1073 by hand: the number 21 is easy but 1073 takes a little more time. Computers have the same problem.

As N becomes large, factoring becomes a hard problem on a classical computer and, in fact, the number of steps required becomes exponential with N. This is how the security of the key arises. If these large numbers can be factored, then an eavesdropper can recover the message sent from Alice to Bob.

Let us examine the RSA protocol which allows Bob to announce publicly a key (the public key) such that Alice can use the key, encrypt a message and send it publicly back to Bob. The protocol only allows Bob to decipher it. The RSA protocol is established as follows.

r It begins by Bob taking two very large prime numbers p and q (of size greater than 101000 ).. He then calcu lates the quantities N = pq (where obviously N > 102000 and pq(N ) = ( p 1) (q 1)). Bob then takes an integer e < N such that the greatest common divisor between e and pq(N ) is one in which case e and pq(N ) are said to be coprime (i.e.

they have no common factors other than 1). Once this is achieved, Bob computes d = e 1 (mod pq(N )). r Bob then sends the public key (e, N) to Alice via a public channel.

r Alice now encodes her plain text message m according to the rule c = m e (mod N ) and sends the result c to Bob. r Once Bob receives the encrypted message c from Alice he computes cd (mod N ) and recovers the plain text message m..

It is interes none for none ting to observe that most of the work in this protocol is done by Bob (not Alice). Alice s main job is to encode the message m she wishes to send to Bob with the public key (e, N). Given that all the communication is done only via a public channel, what is an effective mechanism for an eavesdropper to break this crypto-system Eve can attack this crypto-system quite easily.

To do this Eve must simply solve d = e 1 (mod pq(N )) where she knows only e and N. Effectively this just requires. Applications of entanglement Eve to factor none none N. Once she factors N she obtains p and q and hence can calculate pq(N ). While this may sound like a relatively simple attack, it is not practical to factorize numbers of the size of N > 102000 on a classical computer with current technology in a reasonable time (in fact, to factor the numbers currently used in the RSA encryption system would take a very large supercomputer many, many, years).

However, a recent proposal for a new kind of computer architecture, the quantum computer, opens the possibility for factoring these very large numbers in a short time. The quantum computer, of course, has yet to be realized, but has the potential to revolutionize information processing through its ability to access a vastly larger state space than a classical machine. This, in turn changes the complexity classi cation of how dif cult computational tasks are, so that some tasks which are known to be extremely dif cult (that is, require exponential resources) on a classical machine become easier (need only polynomial resources) on a quantum machine.

The gain comes from using two features of quantum physics: entanglement and quantum interference, allowing massively parallel processing capacity. In 1985 David Deutsch [15] (with a generalization in 1992 co-authored with Richard Jozsa [16]) proved that in quantum mechanics the complexity level of some problems can change dramatically. This paved the way for a number of advances in quantum algorithms, in particular that of Peter Shor s for factoring large numbers.

In 1994, Peter Shor [17] discovered a quantum algorithm that allows one to factor large numbers in polynomial time so that, provided one had a quantum computer, factoring would become essentially as easy as multiplying. Recently, a small-scale NMR quantum machine has factored the number 15 [18]. While this may be a small step forward, it shows the potential of the quantum computer as a demonstration of the implementation of a non-trivial algorithm.

The central point here is that if quantum computing schemes can be scaled up to the point where very large numbers can be factored, RSA crypto-systems become vulnerable to deciphering. The NMR scheme is apparently not scalable to the required degree..

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