1) Alice and Bob use the DiffieâHellman key exchange technique with a common prime q = 1 5 7 and a primitive root a = 5. a. If Alice has a private key XA = 15, find her public key YA. b. If Bob has a private key XB = 27, find his public key YB. c. What is the shared secret key between Alice and Bob?
2) Alice and Bob use the Diffie-Hellman key exchange technique with a common prime q = 2 3 and a primitive root a = 5 . a. If Bob has a public key YB = 1 0 , what is Bobâs private key YB? b. If Alice has a public key YA = 8 , what is the shared key K with Bob? c. Show that 5 is a primitive root of 23.
3) In the DiffieâHellman protocol, each participant selects a secret number x and sends the other participant ax mod q for some public number a. What would happen if the participants sent each other xa for some public number a instead? Give at least one method Alice and Bob could use to agree on a key. Can Eve break your system without finding the secret numbers? Can Eve find the secret numbers?
4) This problem illustrates the point that the DiffieâHellman protocol is not secure without the step where you take the modulus; i.e. the âIndiscrete Log Problemâ is not a hard problem! You are Eve and have captured Alice and Bob and imprisoned them. You overhear the following dialog. Bob: Oh, letâs not bother with the prime in the DiffieâHellman protocol, it will make things easier. Alice: Okay, but we still need a base a to raise things to. How about a = 3? Bob: All right, then my result is 27. Alice: And mine is 243. What is Bobâs private key XB and Aliceâs private key XA? What is their secret combined key? (Donât forget to show your work.)
5) Section 10.1 describes a man-in-the-middle attack on the DiffieâHellman key exchange protocol in which the adversary generates two publicâprivate key pairs for the attack. Could the same attack be accomplished with one pair? Explain.
6) Is (5, 12) a point on the elliptic curve y2 = x 3 + 4 x – 1 over real numbers?
7) This problem performs elliptic curve encryption/decryption using the scheme outlined in Section 10.4. The cryptosystem parameters are E11(1, 7) and G = (3, 2). Bâs private key is nB = 7. a. Find Bâs public key PB. b. A wishes to encrypt the message Pm = (10, 7) and chooses the random value k = 5. Determine the ciphertext Cm. c. Show the calculation by which B recovers Pm from Cm.
8) The following is a first attempt at an elliptic curve signature scheme. We have a global elliptic curve, prime p, and âgeneratorâ G. Alice picks a private signing key XA and forms the public verifying key YA = XAG. To sign a message M: â Alice picks a value k. â Alice sends Bob M, k, and the signature S = M – kXAG. â Bob verifies that M = S + kYA. a. Show that this scheme works. That is, show that the verification process produces an equality if the signature is valid. b. Show that the scheme is unacceptable by describing a simple technique for forging a userâs signature on an arbitrary message.
Sample Solution
on its own, with mounted force, infantry, cannons and an assortment of staff. Napoleon constantly tried to assault an adversary on its flank and guaranteed that in such circumstance âThe triumph is in your graspâ. Napoleonâs flank was assaulted at Leipzig, La Routhier and at Waterloo, and in those three fights he was vanquished. Be that as it may, the splendid arrangement was not secure and was destroyed by the inadequacy of Napoleonâs subordinates Ney and Grouchy. As previously mentioned, one of the primary elements engaged with the associated triumph was the inadequacy of Napoleons Marshals. Napoleons strategies, had they been executed by able fighters, would in all likelihood have worked This, is unmistakably shown in the skirmishes of Quatre Bras and Lingy on the sixteenth June 1815. While the French deliberately won these fights the result of them managed the consequence of the commitment at Waterloo two days after the fact. Napoleon had arranged that while he and 68,000 men would assault the Prussian armed force at Lingy, Marshal Ney and 50,000 men would catch the cross streets at Quatre Bras from the 8000 Prussian protectors and afterward dispatch DâErlonâs corps along the caught Namur street so as to assault the Prussian back and right flank. Incredibly Marshal Ney bombed in the entirety of his goals; he neglected to take the cross streets, he neglected to send dâErlon to Napoleon and he neglected to bind Wellingtonâs powers. Ney had been crushed by Wellington amid the peninsular war and therefore was exceptionally careful and reluctant to submit his powers to the assault. His absence of hostility was a blessing to Wellington as it allowed him to fortify the cross streets with another 30,000 men and 60 guns. When Ney at last propelled his assault, Wellingtonâs men had increased full control of the cross streets, and Nay endured noteworthy misfortunes because of his dithering. The other real goof made by Marshal Ney concerned the aura of dâErlonâs Corps. According to the first arrangement dâErlon had gotten orders from Napoleon to walk to Lingy and fall on the uncovered right flank of the Prussian armed force. Poor correspondence between Napoleons armed force were a contributing component to why Wellington was successful. dâErlonâs Corps walked towards the Emperorâs powers, it got counter requests from Marshal Ney and made a beeline for the Battle of Quatre Bras. This single corps of the French armed force would have turned out to be the conclusive component of the two fights this is on the grounds that the British right flank was on limit a solitary French corp could of gotten through and directed the British armed force driven by Wellington. Be that as it may, it was not able participate in the contention in view of hesitation and poor correspondence from Napoleons marshals [3] Had it come to Lingy, the odds of a definitive triumph over the Prussians would have been higher on the grounds that the Prussians would not have possessed the capacity to help Wellington at Waterloo. The Prussian armed force was beaten however not devastated and components of it were still in great request and ready to assault Placenoit on the French ideal amid the Battle of Waterloo. The corps never battled at Quatre Bras in light of the fact that when it came back to the field it was nightfall and battling had halted. Ney requested the corps to come back to Quatre Bras when it was clearly excessively far away to be of any utilization to him that day and it appears t>
on its own, with mounted force, infantry, cannons and an assortment of staff. Napoleon constantly tried to assault an adversary on its flank and guaranteed that in such circumstance âThe triumph is in your graspâ. Napoleonâs flank was assaulted at Leipzig, La Routhier and at Waterloo, and in those three fights he was vanquished. Be that as it may, the splendid arrangement was not secure and was destroyed by the inadequacy of Napoleonâs subordinates Ney and Grouchy. As previously mentioned, one of the primary elements engaged with the associated triumph was the inadequacy of Napoleons Marshals. Napoleons strategies, had they been executed by able fighters, would in all likelihood have worked This, is unmistakably shown in the skirmishes of Quatre Bras and Lingy on the sixteenth June 1815. While the French deliberately won these fights the result of them managed the consequence of the commitment at Waterloo two days after the fact. Napoleon had arranged that while he and 68,000 men would assault the Prussian armed force at Lingy, Marshal Ney and 50,000 men would catch the cross streets at Quatre Bras from the 8000 Prussian protectors and afterward dispatch DâErlonâs corps along the caught Namur street so as to assault the Prussian back and right flank. Incredibly Marshal Ney bombed in the entirety of his goals; he neglected to take the cross streets, he neglected to send dâErlon to Napoleon and he neglected to bind Wellingtonâs powers. Ney had been crushed by Wellington amid the peninsular war and therefore was exceptionally careful and reluctant to submit his powers to the assault. His absence of hostility was a blessing to Wellington as it allowed him to fortify the cross streets with another 30,000 men and 60 guns. When Ney at last propelled his assault, Wellingtonâs men had increased full control of the cross streets, and Nay endured noteworthy misfortunes because of his dithering. The other real goof made by Marshal Ney concerned the aura of dâErlonâs Corps. According to the first arrangement dâErlon had gotten orders from Napoleon to walk to Lingy and fall on the uncovered right flank of the Prussian armed force. Poor correspondence between Napoleons armed force were a contributing component to why Wellington was successful. dâErlonâs Corps walked towards the Emperorâs powers, it got counter requests from Marshal Ney and made a beeline for the Battle of Quatre Bras. This single corps of the French armed force would have turned out to be the conclusive component of the two fights this is on the grounds that the British right flank was on limit a solitary French corp could of gotten through and directed the British armed force driven by Wellington. Be that as it may, it was not able participate in the contention in view of hesitation and poor correspondence from Napoleons marshals [3] Had it come to Lingy, the odds of a definitive triumph over the Prussians would have been higher on the grounds that the Prussians would not have possessed the capacity to help Wellington at Waterloo. The Prussian armed force was beaten however not devastated and components of it were still in great request and ready to assault Placenoit on the French ideal amid the Battle of Waterloo. The corps never battled at Quatre Bras in light of the fact that when it came back to the field it was nightfall and battling had halted. Ney requested the corps to come back to Quatre Bras when it was clearly excessively far away to be of any utilization to him that day and it appears t>
