Assassination of Martin Luther King Jr.

At 6:01 p.m. on April 4, 1968, Civil Rights leader Dr. Martin Luther King Jr. was hit by a snipers bullet. King had been standing on the balcony in front of his room at the Lorraine Motel in Memphis, Tennessee, when without warning, he was shot. The .30-caliber rifle bullet entered Kings right cheek, traveled through his neck, and finally stopped at his shoulder blade. King was immediately taken to a nearby hospital but was pronounced dead at 7:05 p.m. Violence and controversy followed. In outrage of the murder, many blacks took to the streets across the United States in a massive wave of riots. The FBI investigated the crime, but many believed them partially or fully responsible for the assassination. An escaped convict by the name of James Earl Ray was arrested, but many people,  including some of Martin Luther King Jr.s own family, believe he was innocent. What happened that evening? Dr. Martin Luther King Jr.   When Martin Luther King Jr.  emerged as the leader of the  Montgomery Bus Boycott in 1955, he began a long tenure as the spokesperson for nonviolent protest in the Civil Rights Movement. As a Baptist minister, he was a moral leader to the community. Plus, he was charismatic and had a powerful way of speaking. He was also a man of vision and determination. He never stopped dreaming of what could be. Yet he was a man, not a God. He was most often overworked and overtired and he had a fondness for the private company of women. Though he was the 1964 Nobel Peace Prize winner, he did not have complete control over the Civil Rights Movement. By 1968, violence had edged its way into the movement. Black Panther Party members carried loaded weapons, riots had erupted across the country, and numerous civil rights organizations had taken up the mantra Black Power! Yet Martin Luther King Jr. held strong to his beliefs, even as he saw the Civil Rights Movement being torn in two. Violence is what brought King back to Memphis in April 1968. Striking Sanitation Workers in Memphis On February 12, a total of 1,300  African-American sanitation workers in Memphis went on strike. Though there had been a long history of grievances, the strike began as a response to a January 31 incident in which 22 black sanitation workers were sent home without pay during bad weather while all the white workers remained on the job. When the City of Memphis refused to negotiate with the 1,300 striking workers, King and other civil rights leaders were asked to visit Memphis in support. On Monday, March 18, King managed to fit in a quick stop in Memphis, where he spoke to more than  15,000 who had gathered at Mason Temple. Ten days later, King arrived in Memphis to lead a march in support of the striking workers. Unfortunately, as King led the crowd, a few of the protesters got rowdy and smashed the windows of a storefront. The violence spread and soon countless others had taken up sticks and were breaking windows and looting stores. Police moved in to disperse the crowd. Some of the marchers threw stones at the police. The police responded with tear gas and nightsticks. At least one of the marchers was shot and killed. King was extremely distressed at the violence that had erupted in his own march and became determined not to let violence prevail. He scheduled another march in Memphis for April 8. On April 3, King arrived in Memphis a little later than planned because there had been a bomb threat for his flight before takeoff. That evening, King delivered his Ive Been to the Mountaintop speech to a relatively small crowd that had braved the bad weather to hear King speak. Kings thoughts were obviously on his mortality, for he discussed the plane threat as well as the time he had been stabbed. He concluded the speech with, Well, I dont know what will happen now; weve got some difficult days ahead. But it really doesnt matter with me now, because Ive been to the mountaintop. And I dont mind. Like anybody, I would like to live a long life — longevity has its place. But Im not concerned about that now. I just want to do Gods will. And Hes allowed me to go up to the mountain. And Ive looked over, and Ive seen the Promised Land. I may not get there with you. But I want you to know tonight, that we, as a people will get to the Promised Land. And so Im happy tonight; Im not worried about anything; Im not fearing any man. Mine eyes have seen the glory of the coming of the Lord. After the speech, King went back to the Lorraine Motel to rest. Martin Luther King Jr. Stands on the Lorraine Motel Balcony The Lorraine Motel (now the  National Civil Rights Museum) was a relatively drab, two-story motor inn on Mulberry Street in downtown Memphis. Yet it had become a habit of Martin Luther King and his entourage to stay at the Lorraine Motel when they visited Memphis. On the evening of April 4, 1968, Martin Luther King and his friends were getting dressed to have dinner with Memphis minister Billy Kyles. King was in room 306 on the second floor and hurried to get dressed since they were, as usual, running a bit late. While putting on his shirt and using Magic Shave Powder to shave, King chatted with Ralph Abernathy about an upcoming event. Around 5:30 p.m., Kyles  knocked on their door to hurry them along. The three men joked about what was to be served for dinner. King and Abernathy wanted to confirm that they were going to be served soul food and not something like filet mignon. About half an hour later, Kyles and King stepped out of the motel room onto the balcony (basically the outside walkway that connected all the motels second-story rooms). Abernathy had gone to his room to put on some cologne. Near the car in the parking lot directly below the balcony, waited  James Bevel, Chauncey Eskridge (SCLC lawyer), Jesse Jackson, Hosea Williams, Andrew Young, and Solomon Jones, Jr. (the driver of the loaned white Cadillac). A few remarks were exchanged between the men waiting below and Kyles and King. Jones remarked that King should get a topcoat since it might get cold later; King replied, O.K. Kyles was just a couple of steps down the stairs and Abernathy was still inside the motel room when the shot rang out. Some of the men initially thought it  was a car backfire, but others realized it was a rifle shot. King had fallen to the concrete floor of the balcony with a large, gaping wound covering his right jaw. Martin Luther King Jr. Shot Abernathy ran out of his room to see his dear friend fallen, laying in a puddle of blood. He held Kings head saying, Martin, its all right. Dont worry. This is Ralph. This is Ralph.* Kyles had gone into a motel room to call an ambulance while others encircled King. Marrell McCollough, an undercover Memphis police officer, grabbed a towel and tried to stop the flow of blood. Though King was unresponsive, he was still alive — but only barely. Within 15 minutes of the shot, Martin Luther King arrived at St. Josephs Hospital on a stretcher with an oxygen mask over his face. He had been hit by a .30-06 caliber rifle bullet that had entered his right jaw, then traveled through his neck, severing his spinal cord, and stopped in his shoulder blade. The doctors tried emergency surgery but the wound was too serious. Martin Luther King Jr. was pronounced dead at 7:05 p.m. He was 39 years old. Who Killed Martin Luther King Jr.? Despite many conspiracy theories questioning who was responsible for the assassination of Martin Luther King Jr., most of the evidence points to a single shooter, James Earl Ray. On the morning of April 4, Ray used information from the televised news as well as from a newspaper to discover where King was staying in Memphis. Around 3:30 p.m., Ray, using the name John Willard, rented room 5B in Bessie Brewers run-down rooming house that was located across the street from the Lorraine Motel. Ray then visited the York Arms Company a few blocks away and purchased a pair of binoculars for $41.55 in cash. Returning to the rooming house, Ray readied himself in the communal bathroom, peering out the window, waiting for King to emerge from his hotel room. At 6:01 p.m., Ray shot King, mortally wounding him. Immediately after the shot, Ray quickly placed his rifle, binoculars, radio, and newspaper into a box and covered it with an old, green blanket. Then Ray hurriedly carried the bundle out of the bathroom, down the hall, and down to the first floor. Once outside, Ray dumped his package outside the Canipe Amusement Company and walked swiftly to his car. He then drove away in his white Ford Mustang, just before police arrived. While Ray was driving toward Mississippi, police were starting to put the pieces together. Nearly immediately, the mysterious green bundle was discovered as were several witnesses who had seen someone who they believed to be the new renter of 5B rushing out of the rooming house with the bundle. By comparing fingerprints found on items in the bundle, including those on the rife and binoculars, with those of known fugitives, the FBI discovered they were looking for James Earl Ray. After a two-month international manhunt, Ray was finally captured on June 8 at Londons Heathrow Airport. Ray pleaded guilty and was given a 99-year sentence in prison. Ray died in prison in 1998. * Ralph Abernathy as quoted in Gerald Posner, Killing the Dream  (New York: Random House, 1998) 31. Sources: Garrow, David J.  Bearing the Cross: Martin Luther King, Jr., and the Southern Christian Leadership Conference. New York: William Morrow, 1986. Posner, Gerald.  Killing the Dream: James Earl Ray and the Assassination of Martin Luther King, Jr.  New York: Random House, 1998.

Citizen Soldiers essay - 576 Words

In the book Citizen Soldiers by Stephen E. Ambrose, the title explains mainly what the book is about. The title itself gives you the insight about how the war was fought through the perspective of a regular citizen fighting in the biggest war in history. During the war there were many casualties, as a result more regular citizens were being drafted to go right into battle. In this book Ambrose exemplifies the fact that there were many regular citizens in the war and that they took the situation that they were in and made better of it and overcame it to come home as heroes. The men of the story were ordinary citizens put into an extraordinary situation and came out on top. These men often bonded together through some of the harder times,†¦show more content†¦nbsp;nbsp;nbsp;nbsp;nbsp;The soldiers in the book also had many knew chances to bond together during the war while still concentrating of fighting. â€Å"I never observed any loners on the front line†¦The men automatically paired up. The buddy system worked very well; it provided additional security†¦it provided additional warmth†¦.We were a team, and sharing the adversities of the elements and combat brought us together and created a bond.† Sergeant Egger expressed his observations from the newer soldiers of the company. This quote expresses what was said and what came out of the war due to the fact that everyone had basically the same amount of training because they were normal everyday people from the states and not highly trained soldiers of the U.S Army. This bond was the only thing the soldiers of the army could rely on, the fact that they knew the man standing next to him could and would take a bullet for him and the fact that there was no competitive nature between most of the newer recruits. nbsp;nbsp;nbsp;nbsp;nbsp;The book Citizen Soldiers gave me a bigger perspective on how the life of an average soldier was before, during, and after the war due to the stress of going into the war, the bond they made during the war, and the aftereffects of having made those bonds and sharing the times they had together during the war. That experience could have turned out to beShow MoreRelatedAmerican Patriots: Soldiers, Citizens, And Voters Essay1188 Words   |  5 Pagesopposed those positions were right. Had those right-minded people not openly voiced their disapproval of what the government was doing, our history would not reflect the democratic principles it so publicly espouses. 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Cengage Learning

Questions: 1.explain (do not simply define) meaning.2. Why is word clarity necessary for thought clarity? 3.What are the characteristics of a good word definition? Give an example of a good one and a poor one. 4. Why is it that when we select a word to use, we must not only consider its definition but also its connotation?5. Explain the two stages of critical reading and explain with an example why critical reading is important. Answers: 1. The word meaning is a general term and refers to the idea, which is represented by a phrase and word. It can also be deciphered as the idea, which an individual wants to communicate, by utilizing signs and words. It can also encompass definition and elements. In linguistics, it is communicated through the utilization of language (Mayfield, 2014). 2. Word clarity is necessary for thought clarity because words are more powerful and they invoke a very strong image in the minds of the individuals. Most of the persons do not give words with the credit they deserve, that is why the individuals should not utter without considering the the consequences initially (Mayfield, 2014). 3. The characteristics of a good word definition include the clarity and comprehensibility. A good word definition should be easily understood by the readers and must clearly signify the idea or thought that is being shared to the readers. The example of a good word is epitome and example of a bad word is damn (Mayfield, 2014). 4. When we select a word to use, we must not only consider its definition but also its connotation because connotations are the thoughts, feeling, associations or images that the words waken in the individuals. For e.g. we should use the words like childlike or youthful to describe childlike behavior and should not use the world like kiddish (Mayfield, 2014). 5. The two stages of critical reading encompasses before reading and after reading the text. The individuals need to scrutinize the piece to get a thought of what the text is all about and what is the main argument in it. Critical reading is important because it provides us with the ability to reinterpret and restructure for enhance clarity (Mayfield, 2014). References Mayfield, M. (2014).Thinking for yourself. Cengage Learning.

Jeffrey Santos Essays (1404 words) - Indian Films,

Jeffrey Santos Professor Shrivastava 11/2/2017 2 November 2017 1947 Earth Vs . Cracking Inida Transition from a novel to a film is quite difficult at times especially when it comes to a partition like the one in India in 1947. The main purpose of my essay will be to find the problems and differences with the transition between the two works. Investigating the problem between Bapsi Sidwa's novel Cracking India and its film adaptation Deepa Metha's 1947 Earth by comparing the two together but focusing on Lenny's sexual representation, Ayah's transformation throughout the film and novel and how this causes Ayah's abduction with Ice Candy Man , will be my main target point. I argue that Bapsi Sidwa's Cracking India and Deepa Metha's 1947 Earth discuss the theme of gender and violence. The characters Lenny, Ayah, and Ice Candy Man all were affected by the trauma in some way or other. The film adaptation fails to fully represent the book and has roots to its theme. The film focuses on Lenny's coming of age and how she changes and adapts to her problems with her nanny, Ayah. It induces the violence and partition events as well. In the film , it fails to show us Lenny's struggle with her coming of age due to her mother being extremely abused by her father. Her mother was cheated and physically abused to the fullest by this man. The film also excludes Lenny's introduction to sexual activity when the novel strongly shows us Lenny's sexual situations with Cousin. This is when Ayah makes Lenny extremely upset. Ayah gets lots of looks and respect from men and Lenny sees that and is strongly bothered by it. The novel does a wonderful job with bringing out Lenny's strong attitude towards sexuality and men in general. This chips into how Ayah's and Lenny's relationship basically works and how Lenny feeds off Ayah and learns from how she does things. When looki ng at these stories, they both include the obvious, violence, love, and disloyalty. With the partition being the violence, Lenny and Ayah's opposite sex cravings being love, and Lenny's father cheating on her mother being the disloyalty in this situation. In Metha's film there is another blatant reason for these small stories. Trilogy's such as Fire, Earth, and Water represent a form of violence against the women apart of this area at the time. Fire conceals same-sex female desires that works in context with their morals. Earth is basically the overall violence against the women during this time and Water is the social limitations put in place in Hinduism ( Metha's film). These three simple stories tell a story within themselves that basically explains why love and violence was such a large reason for women struggle existed at this time. In the film it is much easier to decrypt these 3 trilogy's but Metha struggles to implement some of the small details of the bo ok. Obviously the film helps put a visual on Lenny's struggle with this sexual need but the film does leave out a lot of Lenny's sexual awakenings while the book shows us how Lenny used Ayah as a lesson of her own adolescence. Within the entire Partition, both Metha and Sidhwa provide us with Ayah's transformation from a girl that a four year old strives to be to being an abducted rape victim. In the beginning of the novel, our narrator Lenny describes how Ayah gets "covetous glances" from a variety of men-beggars, holy men, hawkers, and so on " ( Sidhwa 3). In the film, Metha uses the Queen's garden scene to show us Ayah's ability to attract all types of men. During this scene Ayah is basically sitting around a bunch of men, having a conversation which all the men look at her with plans on their minds. Once Ayah figures these vibes the men were gives off she puts on her sari and receives physical attractions from these men. Now, in the book Lenny points out that Ice Candy Man tries to put his shoed foot inside of Ayah's sari and says this, " things love to crawl beneath Ayah's sari. Ladybirds, glowworms, Ice Candy Man's toes" ( Sidhwa 19).

History of USS Intrepid World War II Aircraft Carrier

History of USS Intrepid World War II Aircraft Carrier The third Essex-class aircraft carrier built for the US Navy, USS Intrepid (CV-11) entered service in August 1943. Dispatched to the Pacific, it joined in the Allies island-hopping campaign and took part in the Battle of Leyte Gulf and invasion of Okinawa. In the course of the World War II, Intrepid was hit by a Japanese torpedo and three kamikazes. After serving with the occupation forces at the end of the war, the carrier was decommissioned in 1947. Fast Facts: USS Intrepid (CV-11) Nation: United StatesType: Aircraft CarrierShipyard: Newport News Shipbuilding CompanyLaid Down: December 1, 1941Launched: April 26, 1943Commissioned: August 16, 1943Fate: Museum ShipSpecificationsDisplacement: 27,100 tonsLength: 872 ft.Beam: 147 ft., 6 in.Draft: 28 ft., 5 in.Propulsion: 8 Ãâ€" boilers, 4 Ãâ€" Westinghouse geared steam turbines, 4 Ãâ€" shaftsSpeed: 33 knotsRange: 20,000 nautical miles at 15 knotsComplement: 2,600 menArmament4 Ãâ€" twin 5 inch 38 caliber guns4 Ãâ€" single 5 inch 38 caliber guns8 Ãâ€" quadruple 40 mm 56 caliber guns46 Ãâ€" single 20 mm 78 caliber gunsAircraft90-100 aircraft In 1952, Intrepid commenced a modernization program and rejoined the fleet two years later. The next two decades saw it serve in a variety of roles including as a recovery ship for NASA. Between 1966 and 1969, Intrepid conducted combat operations in Southeast Asia during the Vietnam War. Decommissioned in 1974, the carrier has been preserved as a museum ship in New York City. Design Designed in the 1920s and early 1930s, the US Navys Lexington- and Yorktown-class aircraft carriers were built to meet the limitations set forth by the Washington Naval Treaty. This agreement placed restrictions on the tonnage of different types of warships as well as capped each signatorys overall tonnage. These types of limitations were affirmed through the 1930 London Naval Treaty. As global tensions became more severe, Japan and Italy left the agreement in 1936. With the collapse of the treaty system, the US Navy began creating a design for a new, larger class of aircraft carrier and one which drew from the lessons learned from the Yorktown-class. The resulting design was wider and longer as well as included a deck-edge elevator system. This had been used earlier on USS Wasp (CV-7). In addition to carrying a larger air group, the new design mounted a greatly enhanced anti-aircraft armament. Construction Designated the Essex-class, the lead ship, USS Essex (CV-9), was laid down in April 1941. On December 1, work commenced on the carrier that would become USS Yorktown (CV-10) at Newport News Shipbuilding Dry Dock Company. That same day, elsewhere in the yard, workers laid the keel for the third Essex-class carrier, USS Intrepid (CV-11). As the US entered World War II, work progressed on Intrepid and it slid down the ways on April 26, 1943, with the wife of Vice Admiral John Hoover serving as sponsor. Completed that summer, the carrier entered commission on August 16 with Captain Thomas L. Sprague in command. Departing the Chesapeake, the Intrepid completed a shakedown cruise and training in the Caribbean before receiving orders for the Pacific that December. Island Hopping Arriving at Pearl Harbor on January 10, Intrepid commenced preparations for a campaign in the Marshall Islands. Sailing six days later with Essex and USS Cabot (CVL-28), the carrier began raids against Kwajalein on the 29th and supported the invasion of the island. Turning towards Truk as part of Task Force 58, Intrepid took part in Rear Admiral Marc Mitschers highly successful attacks on the Japanese base there. On the night of February 17, as operations against Truk were concluding, the carrier sustained a torpedo hit from a Japanese aircraft which jammed the carriers rudder hard to port. By increasing power to the port propeller and idling the starboard, Sprague was able to keep his ship on course. On February 19, heavy winds forced Intrepid to turn north towards Tokyo. Joking that Right then I wasnt interested in going in that direction, Sprague had his men construct a jury-rig sail to help correct the ships course. With this in place, Intrepid limped back to Pearl Harbor arriving on February 24. After makeshift repairs, Intrepid departed for San Francisco on March 16. Entering the yard at Hunters Point, the carrier underwent full repairs and returned to active duty on June 9. Proceeding to the Marshalls in August, Intrepid began strikes against the Palaus in early September. After a brief raid against the Philippines, the carrier returned to the Palaus to support American forces ashore during the Battle of Peleliu. In the wake of the fighting, Intrepid, sailing as part of Mitschers Fast Carrier Task Force, conducted raids against Formosa and Okinawa in preparation for Allied landings in the Philippines. Supporting the landings on Leyte on October 20, Intrepid became embroiled in the Battle of Leyte Gulf four days later. USS Intrepid (CV-11) during the Battle of Leyte Gulf, 1944. US Naval History and Heritage Command Leyte Gulf and Okinawa Attacking Japanese forces in the Sibuyan Sea on October 24, aircraft from the carrier mounted strikes against enemy warships, including the massive battleship Yamato. The following day, Intrepid and Mitschers other carriers delivered a decisive blow against the Japanese forces off Cape Engaà ±o when they sank four enemy carriers. Remaining around the Philippines, Intrepid sustained heavy damage on November 25 when two kamikazes struck the ship in the course of five minutes. Maintaining power, Intrepid held its station until the resulting fires were extinguished. Ordered to San Francisco for repairs, it arrived on December 20. Repaired by mid-February, Intrepid steamed west to Ulithi and rejoined operations against the Japanese. Sailing north on March 14, it commenced strikes against targets on Kyushu, Japan four days later. This was followed by raids against Japanese warships at Kure before the carrier turned south to cover the invasion of Okinawa. Attacked by enemy aircraft on April 16, Intrepid sustained a kamikaze hit on its flight deck. The fire was soon extinguished and flight operations resumed. Despite this, the carrier was directed to return to San Francisco for repairs. These were completed in late June and by August 6 Intrepids aircraft were mounting raids on Wake Island. Reaching Eniwetok, the carrier learned on August 15 that the Japanese had surrendered. Postwar Years Moving north later in the month, Intrepid served on occupation duty off Japan until December 1945 at which point it returned to San Francisco. Arriving in February 1946, the carrier moved into reserve before being decommissioned on March 22, 1947. Transferred to Norfolk Naval Shipyard on April 9, 1952, Intrepid began an SCB-27C modernization program which altered its armament and updated the carrier to handle jet aircraft. Re-commissioned on October 15, 1954, the carrier embarked on a shakedown cruise to Guantanamo Bay before deploying to the Mediterranean. Over the next seven years, it conducted routine peacetime operations in the Mediterranean and American waters. In 1961, Intrepid was redesignated as an anti-submarine carrier (CVS-11) and underwent a refit to accommodate this role early the following year. USS Intrepid (CV-11) recovers Gemini 3, March 23 ,1965. NASA NASA and Vietnam In May 1962, Intrepid served as the primary recovery vessel for Scott Carpenters Mercury space mission. Landing on May 24, his Aurora 7 capsule was recovered by the carriers helicopters. After three years of routine deployments in the Atlantic, Intrepid reprised its role for NASA and recovered Gus Grissom and John Youngs Gemini 3 capsule on March 23, 1965. After this mission, the carrier entered the yard in New York for a Fleet Rehabilitation and Modernization program. Completed that September, Intrepid deployed to Southeast Asia in April 1966 to take part in the Vietnam War. Over the next three years, the carrier made three deployments to Vietnam before returning home in February 1969. USS Intrepid (CVS-11) in the South China Sea, September 1966. US Naval History and Heritage Command Later Roles Made flagship of Carrier Division 16 with a home port of Naval Air Station Quonset Point, RI, Intrepid operated in the Atlantic. In April 1971, the carrier participated in NATO exercise before beginning a goodwill tour of ports in the Mediterranean and Europe. During this voyage, Intrepid also conducted submarine detection operations in the Baltic and on the edge of the Barents Sea. Similar cruises were conducted each of the following two years. Returning home in early 1974, Intrepid was decommissioned on March 15. Moored at Philadelphia Naval Shipyard, the carrier hosted exhibits during the bicentennial celebrations in 1976. Though the US Navy intended to scrap the carrier, a campaign led by real estate developer Zachary Fisher and the Intrepid Museum Foundation saw it brought to New York City as a museum ship. Opening in 1982 as the Intrepid Sea-Air-Space Museum, the ship remains in this role today.

International cooperation Research Paper Example | Topics and Well Written Essays - 1250 words

International cooperation - Research Paper Example A country therefore may win from exporting one resource but lose a lot on other exportable commodities which may become expensive in nature. It is also pertinent to note that due to higher demand for factors of production in such industries, there is a shortage of labor for other sectors of the economy. This increases the cost of production for other countries and thus making them less competitive in nature. (Corden & Neary, 1989) There are many countries which have been suffering from this phenomenon and the economic mismanagement has resulted into much of the economic damage for them. During 1980s and 1990s, countries like Nigeria faced the consequences of this. Even developed economies like Australia and Canada faced this situation also. Dutch disease is a relationship between the manufacturing sector and increase in exploitation of the natural resources. The overall mechanism is based upon the assumption that strong inflows of foreign currency due to export of natural resources will appreciate the currency of the country and thus may result into nation’s other exports more expensive to export. Though it is normally associated with the natural resources however it can also be referred along with the sharp increase in foreign inflows, foreign direct investment as well as a surge in the prices of natural resources. The core model is based upon non-traded goods sector and two traded goods sector i.e. the booming and the lagging sector. Booming sector is normally based upon the extraction of oil and other natural resources including gold, copper, coffee or cocoa. The lagging sector of the economy is generally based upon the agriculture sector of the economy as well as other manufacturing sector. (Hausman & Rigobon, 2002) There are two possible consequences of this phenomenon i.e. changes in the labor as well as spending. When booming sector starts to pick up, the overall demand for labor

Number theory and RSA Essay Example | Topics and Well Written Essays - 3250 words

Number theory and RSA - Essay Example At this stage, the message is sent to the recipient, for the receiver to get the clear message, the message is decrypted back to original plaintext. The whole process usually rely on a key, this key give a way to encrypt the message by the sender and a way for the listener to figure out what the message says (decrypt). This can be enabled with the use of the old cryptosystem which used the symmetric keys or the newly and more advanced public key cryptosystem. In symmetric keys one had to possess the key that was used to encrypt the message so as to decrypt back the message at the other end; but for the public key cryptosystem knowing the message does not mean one can decrypt the message (Stinson, 2006). For example, the trapdoor function; in this idea, a function ? is for encryption, so, to decrypt one has to find the inverse, ; which is difficult to find, this means that even if you have ? and you understand the output is A, it is still hard to find (A). However, if it happens that you know k, (extra piece information that acts as a lever that lets the inverse fall through by opening the trapdoor), then from ?, A and k one can easily come up with (A), so in this system k is kept secret but the whole message can be displayed or heard by anyone and it does not pose any threat because nobody can decrypt the message. Number Theory In Number theory, theorems have been derived to explain the public key cryptosystem and the key exchange problem (Martin J. Erickson, 2008). Some studies introduce the sage commands to aid in performing basic number theorems such as Euler’s phi function and greatest common divisor. This happens when coding the algorithms and functions, but mathematically it is not a requirement as we will notice in the examples. In definition, integers greater than 1, and their positive divisors are only 1 and itself, then that is a prime number, otherwise if not then it is a composite number (Martin J. Erickson, 2008). The fundamental theorem of arithmetic is the base equation when dealing with number theory is; n The greatest common divisor of k and m, (gcd (k, m)), is the largest number that can divide both k and m. this two positive integers greater than 0 are relatively prime if gcd (k, m) =1. This can be shown by the extended Euclidian algorithm below (T. H. Cormen, 2001), Input: k, m Output: (d, x, y) subject to d=gcd(k,m) and kx + my = d d=k; t=m; x=1; y=0; r=0; s=1; While (t>0) { q = _d/t_ u=x-qr; v=y-qs; w=d-qt x=r; y=s; d=t r=u; s=v; t=w } return (d, x, y) In the above algorithm, these Invariants were used gcd(k,m)=gcd(d,t) kx + my = d kr + ms = t using the Chinese Reminder Theorem(CRT) (Loxton, 1990), we Let m1, m2, ,,, mk be integers subject to gcd(mi, mj) = 1, i j, and †¦ This means that there exist a unique solution modulo m= and this solution is given by (a1, a2,†¦ ak) = (? ) mod m, Where = m /, and = mod This idea can be proven by these four steps; 1. Set M = 2. Find such that + = 1(this is a Bezou t). 3. Note that = 1 if j 4. Therefore, the solution to the system equation becomes: Mentioned above is the Bezout’s Identity, which is a property in the greatest common divisor (Martin J. Erickson, 2008). The Bezout’s identify states that for any two natural numbers k and m, there exist integers x