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The RSA encryption/decryption above appears rather magical Why should it be that by applying the encryption algorithm and then the decryption algorithm, one recovers the original message In order to understand why RSA works, we'll need to perform arithmetic operations using so-called modulo-n arithmetic In modular arithmetic, one performs the usual operations of addition, multiplication and exponentiation However, the result of each operation is replaced by the integer remainder that is left when the result is divided by n We will take n = pq, where p and q are the large prime numbers used in the RSA algorithm Recall that under RSA encryption, a message (represented by an integer), m, is first exponentiated to the power e using modulo-n arithmetic to encrypt Decryption is performed by raising this value to the power d, again using modulo n arithmetic The result of an encryption step, followed by a decryption step is thus (me) d Let's now see what we can say about this quantity We have: (me)d mod n = med mod n Although we're trying to remove some of the "magic" about why RSA works, we'll need to use a rather magical result from number theory here Specifically, we'll need the result that says if p and q are prime, and n = pq, then xy mod is the same as x(y mod



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10.2.1 Joint Strike Fighter (JSF)





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mod n (me)d mod n = m (ed mod But remember that we chose e and d such that ed -1 is exactly divisible (ie, with no remainder) by (p-1)(q1), or equivalently that ed is divisible by (p-1)(q-1) with a reminder of 1, and thus ed mod (p-1)(q-1) = 1 This gives us (me)d mod n = m 1mod n = m ie, that (me)d mod n = m This is the result we were hoping for! By first exponentiating to the power of e (ie, encrypting) and then exponentiating to the power of d (ie, decrypting), we obtain the original value, m Even more remarkable is the fact that if we first exponentiate to the power of d and then exponentiate to the power of e, ie, we reverse the order of encryption and decryption, performing the decryption operation first and then applying the encryption operation, we also obtain the original value, m! (The proof for this result follows the exact same reasoning as above) We will see shortly that this wonderful property of the RSA algorithm, (me)d mod n = m = (md)e mod n will be of great use The security of RSA relies on the fact that there are no known algorithms for quickly factoring a number, in this case the public value n, into the primes p and q If one knew p and q, then given the public value e, one

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could then easily compute the secret key, d On the other hand, it is not know whether or not there exist fast algorithms for factoring a number, and in this sense the security of RSA is not "guaranteed"

The Joint Strike Fighter (JSF) was contested by competing teams from Boeing (incorporating the former McDonnell Douglas) with the X-32 and Lockheed Martin with the X-35. Both teams ew demonstration models with a downselection to an overall winner in 2001. These designs also embody stealth technology. The aircraft are designed to meet the requirements of four Services: the US Air Force; US Navy; US Marines and British Royal Navy. Three main vehicle con gurations are being developed: Conventional Take Off and Landing (CTOL) for the US Air Force Carrier Vehicle (CV) for the US Navy Short Take-Off Vertical Landing (STOVL) for the US Marines and Royal Navy

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