Describe a brute force algorithm for solving the discrete logarithm problem and find the worst-case and average- case time complexity of this algorithm.
Lecture 10 The Port of Boston Act closed the Boston Harbor to all businesses till the Boston people paid for the tea that they had thrown into the harbor. It was part of the Intolerable Acts. The patriot leaders from all the colonies gathered in Philadelphia and held the 1 Continental Congress. The were there to discuss what was going on but didn’t know what to do or how to handle it. The only thing that they decided was they were going to meet again in a year (May 1775). Patrick Henry, a member of the House of Burgess, stood up in the continental congress and said “I am not just a Virginian today, I am also an American.” It said that they were no longer British, that they had a new name. It gave them a different identity. The only other thing that came out of the continental congress first meeting, was they sent out a letter to the colonies to prepare for war. By April of 1775 things had gotten so bad that Boston looked like an occupied military zone. The colonist came up with signs if the British were moving. April 18,1775, the local military commander began to march to Concord, because the patriots had put a cache in concord and they were going to have a big meeting. Sam Adams and John Handcox were supposed to be in attendance. As soon as the British started marching, the lanterns went up and the minute men went to arms. They were ready to do battle in less than a minute, with a certain amount of guns, ammo, and gun powder. The minute men assembled at Lexington. The British got to Lexington and saw 75 men, who opened fired on the British. The British won and marched on to concord. They came into contract again with 200 Americans, who opened fired. The British won again. The Americans ran ahead and took cover in trees, bushes, fences, and would shoot at the British as they walked by. They did this till the British camps. It was known as the battle of Lexington and Concord (The shots heard around the world). This was the start of the American Revolution. The British lost 379 of 1200 soldiers. The Americans lost around 27. The fighting had begun but no call for Independence. It quickly spread across all the colonies. The Second Continental Congress came together and realized that they had a revolution. First they made themselves the leaders of a National Government. They created an army, navy, postal service ( Benjamin Franklin was the first General Post Master), and gave themselves a way to make currency. Second they wrote two documents, The Declaration of Rights and Grievances, that was a laundry list of complaints against Parliament. It described how Parliament had violated their Rights of Englishmen and that Parliament did not have the right to tax the colonies. It was an attempt to explain what Parliament had done. The second document was the Olive Branch Petition, which was sent to King George III begging him to call his soldiers away. They sent Benjamin Franklin and John Adams with the documents, the king turned and walked away when he saw them though. The King knew what was going on and didn’t care. This hurt the Colonist. In January of 1776, a Scottish Immigrant named Thomas Paine wrote a pamphlet named Common Sense. It was a call to arms to the colonists. Paine said that America was young and vigorous while Britain was old and corrupt and that it was only a matter of time before the things that infected Great Britain would infect the colonies. It had such an impact that the Continental Congress set up a committee that began to draft the Declaration of Independence (Thomas Jefferson, John Adams, Benjamin Franklin, and Rodger Sherman). It was passed on July 4,1776. It was when they told Great Britain that they were a new country. America was very disorganized. The colonies were very possessive of their militia. George Washington was given the command of the National Army. The Battle of Bunker Hill in 1776, the Americans had the high ground. Three times the British marched up the hill, each time the Americans slaughtered them. The third time the British won because the Americans ran out of gun powder. The Americans learned that the British were willing to make any sacrifice to bring them back into the British Empire. The British learned that they should not under estimate the American Colonists. For the next two years, Washington tried to train those men to be a professional army. For every one battle he won, he lost three or four. Lecture 11 The British Camp was on the Charleston Peninsula. Patriot militia hid in the hills around the British camps . Commander Gage asked Britian for help and they send General Howel and General Burogoyne. They believed that the Patriots were a bunch of farmers not soldiers. The British were slaughtered, but they ended up taking over Bunker hill when the Americans ran out of gun powder. The Americans learned that the British would do anything to win and the British learned to not under estimate the Americans. The people defending the colonies were militia troops. They were a part time, short term soldier. They also did not operate outside their state. Realizing this the 2 Continental Congress created the Continental Army. They could go to anywhere in the colonies. It was created in 1775. George Washington was named the Commander of the Continental Army. His first army was the Massachusetts Militia. After the Battle at Bunker Hill the British abandoned Charleston Peninsula and sailed to New York. Washington then brought his men to New York. He tried to stand toe to toe with the British and almost slaughtered the American Army. In the Battle of Brooklyn Heights, Washington had his men retreat. But he came up with a strategy to win. He realized that he could not stand toe to toe with the British army. So he would have to be on the defensive. Which meant, that they did not have to win the war but just outlast eh British army. He had four methods. 1) only fight on ground of his choosing where he knows the ground and terrine. 2) When he does go into battle, he would have a limited objective. 3) He can not needlessly bleed his army. 4) always have an escape route. This was the way Washington fought the war because his soldiers were farmers playing soldiers. By the end of 1776 Washington knew that the British were going to their Winter Camp (because there is a limited food supply, clothing, they would be sick, they were dependant on animals). He knew that the British Winter Camps were in New Jersey. On Christmas eve of 1776, Washington crossed the Delaware River and attacked the German mercenaries. It became known as the Battle of Trenton. He took the camp by storm, because the Germans were celebrating and drunk. Washington was not trying to take territory or supplies. He fought the battle, captured the garrison ( 1200germans)and left. Washington did this 67 times that winter. This gave a moral boost to the Americans. That Spring the army was with new recruits. The winter of 17771778 Washington led the army into a frozen hell. They were in Valley Forge, Pennsylvania. The camp was spread out over 20 miles. The temperature dropped, there was lack of food, diseases were spread. Men died by the 100’s because of these conditions. It got so bad that they would boil the leather of their boots and eat it. Washington hung the men that tried to run away. In spring time a prussin soldier name Frederick Von Steubbeal came to the army. He was a professionally trained soldier. Washington made him an officer and he helped train the army to be soldiers. The Battle of Saratoga was the battle that guaranteed that America would win the war. In 1776, New York was thought to be the key to the British victory. The British thought that if they concord New York that they could cut the colonies in half. General Gage would march up from NYC. General Howel would march up from fighting Washington. And General Bargoyne came over from the west. Gage stayed in NYC. Howel stayed to capture Washington. And Bargoyne was a playboy and this was his down fall. His supply lines were so long that he barely got over the Canadian boarder before he came into contact with General Horatio Gates near Saratoga. Gates was shot and Coronal Benedict Arnold took command of the army. He almost led them to victory before he was shot in his sword and it broke his femur. The army started to retreat. He ordered the nurse to put him on his horse. He had himself tied to the horse and rode back into battle. He rallied all the troops and destroyed Burgoyne’s Army. When Kind Louie XVI of France heard about the victory he extended Diplomatic Recognition. This meant that France would give America loans to help them with the army. France also declared war on Great Britain. He sent part of his navy to attack the British ships. A year later he convinced the King of Spain to declare war on the British. This guaranteed American Independence. The British tried to fight the war as cheap as possible because it was so far in dept.