Bob knows that the ElGamal cryptosystem is similar to Diffie-Hellman. To generate the ElGamal keys, Bob selects the cyclic group Z_p with prime p=20876441 and generator g=5 as the public parameters (in decimal). Bob also selects his secret key X between 1 and p-1. The public parameters are p, g, and the value h=g^X mod p, while the private parameter is X. Bob’s ElGamal encryption uses a random nonce Y between 1 and p-1 and for a given message M between 1 and p-1, and it outputs a pair of values (C1,C2), so that C1=g^Y mod p, and C2=M(h^Y) mod p. This pair of values is the ciphertext for M with nonce Y (i.e., the ciphertext is a tuple). Also, the value “h^Y mod p” is called the “shared secret” of Bob.
Bob’s ElGamal decryption receives ciphertext (C1,C2) and his secret key X as input and multiplies C2 with the modular multiplicative inverse of the shared secret. Specifically, if “D=C1^X=g^(XY) mod p” is the shared secret, and “E=D^(-1) mod p” is the modular multiplicative inverse of the shared secret, the plaintext is “M=EC2 mod p”.
[10 points] Bob posts on the Internet the encryption of M=20192834 as (C1,C2)=(9916780, 5260862) using nonce Y1. Can you find Bob’s “shared secret” value? Show all you work.
[15 points] Bob posts another ciphertext (C1,C2)= (7350174, 13786334) for a different message M2 using nonce Y2. What is Bob’s message M2? Implement a program that recovers Bob’s message M2.
[5 points] Bob notices that ElGamal encryption is malleable. If the encryption of M3=12345 is (C1,C2)= (8698838, 17288353), what is the encryption of M4=382695 (all numbers are decimal)? Show all your work.
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Just utilizing a topical concentrate of nutrient An, or taking nutrient An orally, can add to your skin's wellbeing and in general appearance" ("5 Home Remedies for Stretch Marks"). Consequently, you ought to be vigilant for items and nourishments that contain adequate measures of Vitamin A to make your skin return back to typical. Another amazement is that sugar can help in disposing of stretch imprints. In spite of the fact that this substance is generally thought of as something destructive to our wellbeing, utilizing a sugar clean on the imprints can be of extraordinary advantage. As indicated by Perfect Body Mate, "Blend two tablespoons of common white sugar with some almond oil. 1. Apply it on zones of stretch imprints for a couple of moments. 2. Let your skin assimilate for 5 minutes. 3. Wash it off with warm water or scrub down. Proceed with this for at any rate a month to see your stretch imprints blur away" ("17 Simple Ways to Get Rid of Stretch Marks (Fast)"). 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The following guide sounds somewhat specialized, yet look into has indicated that hyaluronic corrosive can help diminish stretch stamps and even forestall them. It has more proof that aloe vera as an answer. As indicated by The American Academy of Dermatology, "In two enormous examinations, applying hyaluronic corrosive to early stretch imprints made the stretch stamps less observable. Analysts have discovered that items containing centella or hyaluronic corrosive may help forestall stretch imprints. Centella is a herb, and our skin normally contains hyaluronic corrosive" ("Stretch Marks: Why They Appear and How to Get Rid of Them"). Along these lines, this compound may be the most experimentally demonstrated method for evacuating stretch imprints. A home cure that has been famous for an assortment of skin issues for ages is coconut oil. As per Healthline, "Since stretch imprints are scarring from skin harm, coconut oil may help recuperate the appearance rapidly. Coconut oil has been read for its mending properties and was found in rodents to diminish the time it takes for skin wounds to recuperate. Applying virgin coconut oil to your stretch denotes every day may remove a portion of their red appearance" ("5 Home Remedies for Stretch Marks). It is a protected and common guide for disposing of stretch imprints. These arrangements are promptly accessible to individuals to evacuate stretch imprints: taking Vitamin An, utilizing a sugar scour, utilizing aloe vera, ingesting hyaluronic corrosive through containers, and utilizing coconut oil. Every one of these techniques and more are practical, however it is ideal to check with your primary care physician about the best ones as indicated by your condition, sensitivities, and restorative past. Works Cited "5 Home Remedies for Stretch Marks." Healthline, Healthline Media, www.healthline.com/wellbeing/home-solutions for stretch-marks#coconut-oil. "17 Simple Ways to Get Rid of Stretch Marks (Fast)." Perfect Body Mate, 9 Apr. 2019, perfectbodymate.com/stretch-mark-evacuation/#sugar. "Stretch Marks: Why They Appear and How to Get Rid of Them." Stretch Marks: Why They Appear and How to Get Rid of Them | American Academ>GET ANSWER