COMPUTER METHODS FOR DISCRETE LOGARITHM PROBLEM A CRYPTOGRAPHIC PERSPECTIVE

COMPUTER METHODS FOR DISCRETE LOGARITHM

PROBLEM:A CRYPTOGRAPHIC PERSPECTIVE

SHI BAI

1.Research Proposal

Compution of discrete logarithm is of great interest because of its relevance to

public key cryptography.It is believed to be a hard problem because no polynomial time solution has been found.One of the generic algorithm in computing discrete logarithm is Pollard rho algorithm (and its derivatives).The algorithm depend on the ”birthday paradox”idea and makes a pseudo-random walk in ?nite abelian group.If the group is of order n,the algorithm has expected running time O (√n ),provided the pseudo-random walk behaves like a genuine random walk.Further-more,storage requirements are minimal.In some cases such as the group of an elliptic curve,the algorithm is the fastest known method currently.

Very little has been proved about the algorithms because the walks performed

are not truly random.The project will involve theoretical analysis of characteris-tics of the iterating functions which generate the pseudo-random walks.Empirical experiments will also be conducted to examine the performance of pseudo-random walks comparatively.Rivisits of current algorithms will be the basis and starting point for the project.Implementation will be conducted by GNU C ++compiler with multiprecision library.

The aim of this project is to investigate how good the random walk assumption

is for each iterating function.This will help propose and choose a more e?cient iterating function for computing discrete logarithm problem.It will be useful in cryptographic applications where the group of an elliptive curve is involved.The potential improvements will also be applicable to compute the structure of generic groups.

Date :March 16,2007.

1

2SHI BAI

2.Research Scope and Review Plan

The Pollard rho method and its derivatives for computing discrete logarithm

problems will be concentrated through the project.We hope to understand a num-

ber of things such as,

First of all,the conjectured optimal bound of Pollards rho algorithms is O (√n )

under the genuine random assumption.However,the performance of original Pol-

lard’s rho algorithm is slower than that in the truly random case.It has been

showed that the walk originally used by Pollard in discrete logarithm problems

does not behave like a random walk [1].In order to show how good the random

walk assumption is,we may investigate the performance empirically.We could com-

pute the the sum of the period and the length of the non-periodic part of the walk

experimentally for various primes p .Under the truly random assumption,sum 2

should have a certain exponential distribution.Thus we could show how good or

how randomly the Pollard rho algorithm does by investigating the distribution.

Furthermore,for Pollards rho method,the iteration function is used to de?ne a

pseudo-random sequence.The running time depends on the randomness of the ran-

dom walk.Di?erent iteration function would take various steps of attempt.Thus

a comparison of di?erent generating functions for sequence would help in analyzing

the performance.There are some paper in this ?eld such as [2]and [1].Thus We

would like look into the performance of di?erent iteration functions and the inves-

tigation would also help analyze the randomness of function.In addition,there is

a rigorous proof [3]shows that the classical Pollards rho algorithm is bounded by O (√n (log n )3),which is near to the conjectured optimal bound.This may help

build a theoretical foundation for analyzing the performance of the iterative func-

tion for all above investigations.Some papers also give bene?cial results from other

perspective such as Markov Chain [4].

Finally,it would be attractive and exciting to develop some more e?cient pseudo-

random function.While due to the time limitation,this should be served as an

optional target.The ?nal outcome of this project is not only to study and research

in the particular ?eld,but also learn and develop a methodology in future research

and work.

COMPUTER METHODS FOR DISCRETE LOGARITHM PROBLEM:A CRYPTOGRAPHIC PERSPECTIVE3

3.Detailed Time Table

This links to my detailed time table in form of Gantt Chart,which is also an assignment of the course ENGN8103Research Methods and Technique in

this semester.The Gantt Chart will be updated.

https://www.wendangku.net/doc/3515568448.html,/schedule

References

1.TESKE,E,On random walks for pollards rho method.Mathematics of Computation,2001,

70,234,809—825.

2.BRENT,R.P,An improved Monte Carlo factorization algorithm.BIT Numerical Mathemat-

ics20,2,176—184,1980.

https://www.wendangku.net/doc/3515568448.html,LER,S.D.AND VENKATESAN,R,Spectral analysis of pollard rho collisions.2006,

Algorithmic Number Theory Symposium(ANTS VII).

4.R.Montenegro AND P.Tetali,Mathematical aspects of mixing times in Markov chains.

Foundations and Trends in Theoretical Computer Science,2006,Volume1,Issue3:237—

354

College of Engineering and Computer Science,The Australian National University, Canberra ACT,Australian.

E-mail address:shih.bai@https://www.wendangku.net/doc/3515568448.html,

离散数学(第五版)清华大学出版社第2章习题解答

离散数学（第五版）清华大学出版社第2章习题解答 2.1 本题没有给出个体域,因而使用全总个体域. (1) 令F(x):x是鸟 G(x):x会飞翔. 命题符号化为 ?x(F(x)→G(x)). (2)令F(x):x为人. G(x):x爱吃糖 命题符号化为 ??x(F(x)→G(x)) 或者 ?x(F(x)∧?G(x)) (3)令F(x):x为人. G(x):x爱看小说. 命题符号化为 ?x(F(x)∧G(x)). (4) F(x):x为人. G(x):x爱看电视. 命题符号化为 ??x(F(x)∧?G(x)). 分析1°如果没指出要求什么样的个体域，就使用全总个休域，使用全总个体域时，往往要使用特性谓词。（1）-（4）中的F(x)都是特性谓词。 2°初学者经常犯的错误是，将类似于（1）中的命题符号化为 27 ?x(F(x)∧G(x)) 即用合取联结词取代蕴含联结词，这是万万不可的。将（1）中命题叙述得更透彻些，是说“对于宇宙间的一切事物百言，如果它是鸟，则它会飞翔。”因而符号化应该使用联结词→而不能使用∧。若使用∧，使（1）中命题变成了“宇宙间的一切事物都是鸟并且都会飞翔。”这显然改变了原命题的意义。

3°（2）与（4）中两种符号化公式是等值的，请读者正确的使用量词否定等值式，证明（2），（4）中两公式各为等值的。 2.2 (1)d (a),(b),(c)中均符号化为 ?xF(x) 其中F(x):(x+1)2=x2+2x+1,此命题在(a),(b),(c)中均为真命题。 （2）在(a),(b),(c)中均符号化为 ?xG(x) 其中G(x):x+2=0，此命题在（a）中为假命题，在(b)(c)中均为真命题。 （3）在(a),(b),(c)中均符号化为 ?xH(x) 其中H(x):5x=1.此命题在(a),(b)中均为假命题，在(c)中为真命题。 分析1°命题的真值与个体域有关。 2°有的命题在不同个体域中，符号化的形式不同，考虑命题 “人都呼吸”。 在个体域为人类集合时，应符号化为 ?xF(x) 这里，F(x):x呼吸，没有引入特性谓词。 在个体域为全总个体域时，应符号化为 ?x(F(x)→G(x)) 这里，F(x):x为人，且F(x)为特性谓词。G(x):x呼吸。 28 2．3 因题目中未给出个体域，因而应采用全总个体域。 （1）令：F(x):x是大学生，G(x):x是文科生，H(x):x是理科生，命题符号化为?x(F(x)→(G(x)∨H(x)) （2）令F(x):x是人，G(y):y是化，H(x):x喜欢，命题符号化为 ?x(F(x)∧?y(G(y)→H(x,y))) （3）令F(x):x是人，G(x):x犯错误，命题符号化为 ??x(F(x)∧?G(x)), 或另一种等值的形式为 ?x(F(x)→G(x)

《离散数学》清华大学出版社 前四章小测验

一、判断题 (正确的在括号内填写“√”，错误的写“×”) （ ）1、“只有天下大雨，他才乘班车上班”与“除非天下大雨，否则他不乘班车上班”所表达的逻辑关系是一样的。 （ ）2、公式B A ,含相同的命题变项,若B A ∨是重言式,则A 与B 都是重言式。 （ ）3、在命题逻辑中，任何命题公式的主合取范式都是存在的，并且是惟一的。 （ ）4、永真式的主合取范式是1，矛盾式的主析取范式是0。 （ ）5、同一个谓词公式，在不同个体域中，真值不一定相同。 二、单项选择题 1、下列命题是复合命题的是（ ） A 、黄色和蓝色可以调配成绿色； B 、李辛与李未是兄弟； C 、黄色和蓝色都是常用颜色； D 、张辉与王力是同学 ； 2、设p:天下大雨，q:小王乘公共汽车上班，命题“只有天下大雨，小王才乘公共汽车上班”的符号化形式为（ ） A 、p →q ； B 、q →p ； C 、p →┐q ； D 、┐p →q ； 3、公式()p q r ∧?∨的公式类型为（ ） A 、重言式； B 、矛盾式； C 、非重言式的可满足式； 4、下列联结词集是联结词完备集的是（ ） A 、{,,}∧∨→； B 、{,}?→； C 、{,,}∨→?； D 、{,,,}∧∨→?； 5、设解释I 如下：个体域D={a,b},F(a,a)=(b,b)=0,F(a,b)=F(b,a)=1，在解释I 下，下列公式中真值为1的是（ ） A 、(,)x yF x y ??； B 、(,)x yF x y ??； C 、(,)x yF x y ??； D 、(,)y xF x y ??； 三、填空题 1、公式)(q p ??与)()(q p q p ∧?∨?∧共同的成真赋值为 。 2、设命题公式A 为含命题变项,p q r ,的重言式，则公式(())A p q r ∨∧→的类型为 。

离散数学课后习题答案 (邱学绍)

第一章 命题逻辑 习题1.11．解 ⑴不是陈述句，所以不是命题。 ⑵x 取值不确定，所以不是命题。 ⑶问句，不是陈述句，所以不是命题。 ⑷惊叹句，不是陈述句，所以不是命题。 ⑸是命题，真值由具体情况确定。 ⑹是命题，真值由具体情况确定。 ⑺是真命题。 ⑻是悖论，所以不是命题。 ⑼是假命题。 2．解 ⑴是复合命题。设p ：他们明天去百货公司；q ：他们后天去百货公司。命题符号化为q p ∨。 ⑵是疑问句，所以不是命题。 ⑶是悖论，所以不是命题。 ⑷是原子命题。 ⑸是复合命题。设p ：王海在学习；q ：李春在学习。命题符号化为p ∧q 。 ⑹是复合命题。设p ：你努力学习；q ：你一定能取得优异成绩。p →q 。 ⑺不是命题。 ⑻不是命题 ⑼。是复合命题。设p ：王海是女孩子。命题符号化为：?p 。 3．解 ⑴如果李春迟到了，那么他错过考试。 ⑵要么李春迟到了，要么李春错过了考试，要么李春通过了考试。 ⑶李春错过考试当且仅当他迟到了。 ⑷如果李春迟到了并且错过了考试，那么他没有通过考试。 4．解 ⑴?p →(q ∨r )。⑵p →q 。⑶q →p 。⑷q → p 。 习题1.2 1．解 ⑴是1层公式。 ⑵不是公式。 ⑶一层： p ∨q ，?p 二层：?p ?q 所以，)()(q p q p ??→∨是3层公式。 ⑷不是公式。 ⑸(p →q )∧?(?q ?( q →?r ))是5层公式，这是因为 一层：p →q ，?q ，?r 二层：q →?r 三层：?q ?( q →?r ) 四层：?(?q ?( q →?r )) 2．解 ⑴A =(p ∨q )∧q 是2层公式。真值表如表2-1所示： 表2-1 ⑵p q p q A →→∧= )(是3层公式。真值表如表2-2所示：

高中英语必修2课文(人教版)

高中英语必修2（人教版） Unit1 Cultural relics IN SEARCH OF THE AMBER ROOM Frederick William I, the King of Prussia, could never have imagined that his greatest gift to the Russian people would have such an amazing history. This gift was the Amber Room, which was given this name because several tons of amber were used to make it. The amber which was selected had a beautiful yellow-brown colour like honey. The design of the room was in the fancy style popular in those days. It was also a treasure decorated with gold and jewels, which took the country's best artists about ten years to make. In fact, the room was not made to be a gift. It was designed for the palace of Frederick I. However, the next King of Prussia, Frederick William I, to whom the amber room belonged, decided not to keep it. In 1716 he gave it to Peter the Great. In return, the Czar sent him a troop of his best soldiers. So the Amber Room became part of the Czar's winter palace in St Petersburg. About four metres long, the room served as a small reception hall for important visitors.

清华大学-计算机专业-培养计划

一、培养目标 信息科学技术学院(以下简称信息学院)本科培养方案面向电子信息科学与技术、计算机科学与技术、自动化、微电子学、示范性软件学院的计算机软件等五个专业，从2003级开始实行多学科交叉背景下、通识教育基础上的宽口径专业教育，构建具有各专业共性基础的学院平台课程体系以及具有一定特长的专业核心课程体系，强调对学生进行基本理论、基础知识、基本能力(技能)以及健全人格、综合素质和创新精神培养，为学生提供增强基础、选择专业的机制，培养基础厚、专业面宽、具有自主学习能力的复合型人才。 从2011级开始，信息学院对培养方案进行了全面修订，进一步将学科交叉范围扩大到专业核心课程体系，为学生提供更加灵活的选课机制和更加宽广的专业空间；并将继续深入研究和不断改进课程内容和教学方法，加强实践环节，更好地培养适应时代要求的信息科学技术专业人才。 信息学院致力于为学生全面参与教育教学、科学研究、文化艺术、社会服务等活动创造条件，提倡学生在参与中发现自己的能力和兴趣，最大限度地发展自己的智力和潜能，鼓励学生敢于面对挑战、不断探索、努力创造、追求卓越，并提供一种基础和环境，促使学生养成独立工作的能力和终身学习的习惯。 二、基本要求 信息学院各专业通过各种教育教学活动发展学生个性，培养学生具有健全人格；具有成为高素质、高层次、多样化、创造性人才所具备的人文精神以及人文、社科方面的背景知识；具有国际化视野；具有创新精神；具有提出、解决带有挑战性问题的能力。具有进行有效的交流与团队合作的能力；在信息科学技术领域掌握扎实的基础理论、相关领域基础理论和专门知识及基本技能，具有在相关领域跟踪、发展新理论、新知识、新技术的能力，能从事相关领域的科学研究、技术开发、教育和管理等工作。 电子信息科学与技术专业的本科生运用所掌握的理论知识和技能，从事信号获取、处理和应用，通信及系统和网络，模拟及数字集成电路设计和应用，微波及电磁技术理论、信号与信息处理的新型电子材料、器件和系统，包括信息光电子和光子器件、微纳电子器件、微光机电系统、大规模集成电路和电子信息系统芯片的理论和应用等方面的科研、开发与教育工作。 微电子学专业的本科生运用所掌握的理论知识和技能，从事大规模模拟及数字集成电路设计和应用，工艺开发，EDA工具开发，新型电子材料、微纳电子器件和系统，量子信息和电子信息系统的理论和应用等方面的科研、开发与教育工作。培养基础扎实，创新能力突出，有国际视野的微纳电子专业人才。 计算机科学与技术专业的本科生运用所掌握的理论知识和技能，从事计算机科学理论、计算机系统结构、计算机网络、计算机软件及计算机应用技术等方面的科研、开发与教育工作。 自动化专业的本科生运用所掌握的理论知识和技能，从事国民经济、国防和科研各部门的运动控制、过程控制、机器人智能控制、导航制导与控制，现代集成制造系统、模式识别与智能系统、生物信息学、人工智能与神经网络、系统工程理论与实践、新型传感器、电子与自动检测系统、复杂网络与计算机应用系统等领域的科学研究、技术开发、教育及管理等工作。 计算机软件专业的本科毕业生应该具备扎实的软件理论和软件工程专业基础知识，具有良好的工具使用与实验能力、软件分析与开发能力、过程控制与管理能力、团队协作与沟通能力。 三、学制与学位授予

人教版高一英语必修二英语课文原文

Frederick William Ⅰ，the King of Prussia , could never have imagined that his greatest gift to the Russian people would have such an amazing history . This gift was the Amber Room , which was given this name because several tons of amber were used to make it . The amber which was selected had a beautiful yellow-brown colour like honey . The design of the room was in the fancy style popular in those days . It was also a treasure decorated with gold and jewels , which took the country's best artists about ten years to make . In fact , the room was not made to be a gift . It was designed for the palace of Frederick Ⅰ. However, the next King of Prussia , Frederick William Ⅰ,to whom the amber room belonged, decided not to keep it. In 1716 he gave it to Peter the Great. In return , the Czar sent him a troop of his best soldiers. So the Amber Room because part of the Czar's winter palace in St Petersburg.About four metres long, the room served as a small reception hall for important visitors . Later,Catherine Ⅱhad the Amber Room moved to a palace outside St Petersburg where she spent her summers. She told her artists to add more details to it .In 1770 the room was completed the way she wanted . Almost six hundred candles lit the room ,and its mirrors and pictures shone like gold. Sadly , although the Amber Room was considered one of the wonders of the world , it is now missing . In September 1941, the Nazi army was near St Petersburg . This was a time when the two countries were at war . Before the Nazis could get to the summer palace , the Russians were able to remove some furniture and small art objects from the Amber Room . However , some of the Nazis secretly stole the room itself . In less than two days 100,000 pieces were put inside twenty-seven woooden boxs . There is no doubt that the boxs were then put on a train for Konigsberg, which was at that time a German city on the Baltic Sea . After that, what happened to the Amber Room remains a mystery . Recently , the Russians and Germans have built a new Amber Room at the summer palace . By studying old photos of the former Amber Room , they have made the new one look like the old one .In 2003 it was ready for the people of St Petersburg when they celebrated the 300th birthday of their city . A FACT OR AN OPINION? What is a fact? Is it something that people believe? No. A fact is anything that can be proved. For example, it can be proved that China has more people than any other country in the world. This is a fact. Then what is an opinion? An opinion is what someone believes is true but has not been proved. So an opinion is not good evidence in a trial. For example, it is an opinion if you say “Cats are better pets than dogs”. It may be true, but it is difficult to prove. Some people may not agree with this opinion but they also cannot prove that they are right. In a trial, a judge must decide which eyewit nesses to believe and which not to believe. The judge does not consider what each eyewitness looks like or where that person lives or works. He/she only cares about whether the eyewitness has given true information, which must be facts rather than opinions. This kind of information is called evidence. Unit 2 AN INTERVIEW Pausanias, who was a Greek writer about 2,000 years ago, has come on a magical journey on March 18th 2007 to find out about the present-day Olympic Games. He is now interviewing Li Yan, a volunteer for the 2008 Olympic Games.

离散数学复习题

选择题 1.设P :他在听音乐，Q :他学习，将命题“他在学习或在听音乐”符号化正确的是（ ） A.P →Q B.P ∧Q C.P ∨Q D.Q →?P 2.下列命题公式不是．． 永真式的是( ) A.()P Q P →→ B.()P Q →∨P C.P ?∨()Q P → D.()P Q P →→ 3.下列等价式正确的是（ ） A.()()()()x A x x A x ????? B.()()()(())A x B x x A B x →???→ C.()(())()()x A x B x A x B ?→??→ D.()(())()()x A x B x A x B ?→??→ 4.设A(x):x 是鸟，B(x):x 会飞，命题“有的鸟不会飞”符号化为（ ） A.()(()x A x ??∧())B x B.()(()x A x ??∧())B x C.()(()())x A x B x ??→ D.()(()())x A x B x ??→ 5.设X ={,{},{,}}a a ??，则下列陈述正确的是（ ） A.a X ∈ B.{,}a X ?? C.{{,}}a X ?? D.{}X ?∈ 6.设A B B = ,则有（ ） A.A B A = B.A B -=? C.A B B = D.A B ? 7.设A =｛a ,{b , c }｝,则其幂集P （A ）的元素总个数为（ ） A.3 B.4 C.6 D.8 8.在整数集Z 上，下列定义的运算满足结合律的是（ ） A.1a b b *=+ B.1a b a *=- C.1a b ab *=- D.1a b a b *=++ 9.设是群，则下列陈述不正确．．． 的是（ ） A.11()a a --= B.111()ab a b ---= C.n m n m a a a += D.11()n n a ba a b a --= 10.设:,:f X Y g Y Z →→是函数，则下列陈述正确的是（ ） A.若f 不是入射的，则g f 不是入射的 B.若g 是入射的，则g f 也是入射的 C.若f 是入射的，则g f 也是入射的 D.若g f 不是入射的，则f 也不是入射的 11.设简单图G 所有结点的度数之和为36，由G 的边数为（ ） A.6 B.9 C.12 D.18

英语必修二课文原文

人教版新课标必修2u n i t1课文原文和翻译w o r d格式 IN SEARCH OF THE AMBER ROOM Frederick William Ⅰ，the King of Prussia , could never have imagined that his greatest gift to the Russian people would have such an amazing history . This gift was the Amber Room , which was given this name because several tons of amber were used to make it . The amber which was selected had a beautiful yellow-brown colour like honey . The design of the room was in the fancy style popular in those days . It was also a treasure decorated with gold and jewels , which took the country's best artists about ten years to make . In fact , the room was not made to be a gift . It was designed for the palace of Frederick Ⅰ. However, the next King of Prussia , Frederick William Ⅰ,to whom the amber room belonged, decided not to keep it. In 1716 he gave it to Peter the Great. In return , the Czar sent him a troop of his best soldiers. So the Amber Room because part of the Czar's winter palace in St four metres long, the room served as a small reception hall for important visitors . Later,Catherine Ⅱhad the Amber Room moved to a palace outside St Petersburg where she spent her summers. She told her artists to add more details to it .In 1770 the room was completed the way she wanted . Almost six hundred candles lit the room ,and its mirrors and pictures shone like gold. Sadly , although the Amber Room was considered one of the wonders of the world , it is now missing . In September 1941, the Nazi army was near St Petersburg . This was a time when the two countries were at war . Before the Nazis could get to the summer palace , the Russians were able to remove some furniture and small art objects from the Amber Room . However , some of the Nazis secretly stole the room itself . In less than two days 100,000 pieces were put inside twenty-seven woooden boxs . There is no doubt that the boxs were then put on a train for Konigsberg, which was at that time a German city on the Baltic Sea . After that, what happened to the Amber Room remains a mystery . Recently , the Russians and Germans have built a new Amber Room at the summer palace . By studying old photos of the former Amber Room , they have made the new one look like the old one .In 2003 it was ready for the people of St Petersburg when they celebrated the 300th birthday of their city . A FACT OR AN OPINION What is a fact Is it something that people believe No. A fact is anything that can be proved. For example, it can be proved that China has more people than any other country in the world. This is a fact. Then what is an opinion An opinion is what someone believes is true but has not been proved. So an opinion is not good evidence in a trial. For example, it is an opinion if you say “Cats are better pets than dogs”. It may be true, but it is difficult to prove. Some peo ple may not agree with this opinion but they also cannot prove that they are right. In a trial, a judge must decide which eyewitnesses to believe and which not to believe. The judge does not consider what each eyewitness looks like or where that person lives or works. He/she only cares about whether the eyewitness has given true information, which must be facts rather than opinions. This kind of information is called evidence. 人教版新课标必修2 unit2 课文原文word格式 AN INTERVIEW Pausanias, who was a Greek writer about 2,000 years ago, has come on a magical journey on March 18th 2007 to find out about the present-day Olympic Games. He is now interviewing Li Yan, a volunteer for the 2008 Olympic Games.

必修2 人教版高中英语课文原文和翻译

实用标准 Book2 Unit1 Cultural relics IN SEARCH OF THE AMBER ROOM 寻找琥珀屋 Frederick William I, the King of Prussia, could never have imagined that his greatest gift to the Russian people would have such an amazing history. This gift was the Amber Room, which was given this name because several tons of amber were used to make it. The amber which was selected had a beautiful yellow-brown colour like honey. The design of the room was in the fancy style popular in those days. It was also a treasure decorated with gold and jewels, which took the country's best artists about ten years to make. 普鲁士国王腓特烈·威廉一世绝不可能想到他送给俄罗斯人民的厚礼会有一段令人惊讶的历史。 这件礼物就是琥珀屋,它之所以有这个名字,是因为造这间房子用了近几吨琥珀, 被选择的琥珀色彩艳丽,呈黄褐色,像蜜一样。屋子的设计当时流行的极富艺术表现力的建筑风格。琥珀屋这件珍品还镶嵌著黄金和珠宝,全国最优秀的艺术家用 了是年的时间才完成它。 In fact, the room was not made to be a gift. It was designed for the palace of Frederick I. However, the next King of Prussia, Frederick William I, to whom the amber room belonged, decided not to keep it. In 1716 he gave it to Peter the Great. In return, the Czar sent him a troop of his best soldiers. So the Amber Room became part of the Czar's winter palace in St Petersburg. About four metres long, the room served as a small reception hall for important visitors. 事实上,琥珀屋并不是作为礼物而建造的。它是作为腓烈特一世的宫殿而建造的。然而,下一位普鲁士国王,腓烈特·威廉一世,这个琥珀屋的主人却决定不再要它了。1716年,他把它送给了彼得大帝。作为回馈,沙皇则送给他一队自己最好的士兵。所以,琥珀屋就成了沙皇在圣彼得堡东宫的一部分。琥珀屋长约4米,被用作接待 重要来宾的小型会客室。 Later, Catherine II had the Amber Room moved to a palace outside St Petersburg where she spent her summers. She told her artists to add more details to it. In 1770 the room was completed the way she wanted. Almost six hundred candles lit the room, and its mirrors and pictures shone like gold. Sadly, although the Amber Room was considered one of the wonders of the world, it is now missing. 后来、叶卡捷琳娜二世派人把琥珀屋搬到了圣彼得堡郊外避暑的宫殿中。她让艺术家们给它增添了更多的装饰。1770年,这间琥珀屋按她要求的方式完工了。将 近600支蜡烛照亮了这个房间,里面的镜子和图画就像金子一样闪闪发光。不幸 的是,虽然琥珀屋被认为是世界奇迹之一,可是现在它却下落不明。 In September 1941, the Nazi army was near St Petersburg. This was a time when the two countries were at war. Before the Nazis could get to the summer palace, the Russians were able to remove some furniture and small art

清华大学计算机系培养方案一

清华大学计算机系培养方案一

信息科学技术学院 本科指导性教学计划 第一年 课程编号课程名称学分周学时考核方式说明及主要先修课 12090043 军事理论与技能训练3 3 考查 秋季学期 课程编号课程名称学分周学时考核方式说明及主要先修课 107 1 体育(1) 1 2 考查 10610183 思想道德修养与法律基础3 2 考查 10640532 英语（1） 2 2 考查 10420874 一元微积分 4 4 考试 10420904 几何与代数(1) 4 4 考试 0412 工程图学基础 2 2 考试 2选1 20240023 离散数学(2) 3 3 考试 程序设计课组 3 3 考试选１门，详见附录2 30210041 信息科学技术概论 1 1 考查 文化素质选修课≥1 1

合计≥21 注：计算机科学与技术专业必修“离散数学(2)”，其它专业必修“工程图学基础”。 春季学期 课程编号课程名称学分周学时考核方式说明及主要先修课 107 1 体育(2) 1 2 考查 10610193 中国近现代史纲要 3 2 考试 10640682 英语（2） 2 2 考查 10420884 多元微积分 4 4 考试先修一元微积分 10421002 几何与代数(2) 2 2 考试 大学物理课组1 4 4 考试先修一元微积分 20220214 电路原理 4 4 考试 20220221 电路原理实验 1 1 考查 合计≥21 夏季学期 课程编号课程名称学分周学时考核方式说明及 主要先修课 - 3 -

- 4 - 21510192 电子工艺实习（集中） 2 考查 2周 程序训练课组 2 考查 3周 合计： 4 第二年 秋季学期 课程编号 课程名称 学分 周学时 考核方式 说明及 主要先修课 107 1 体育(3) 1 2 考查 10610204 马克思主义基本原理 4 3 考试 10420892 高等微积分B 2 2 考试 10420252 复变函数引论 2 2 考试 304 3 复分析 3 3 考试 大学物理课组2 4 4 考试 见附录2 10430782 物理实验A(1) 2 2 考查 10430801 物理实验B(1) 1 1 考查 电子基础课组１ 3 3 考试 电子基础课组２ 3 3 考试 21550012 电子技术实验 2 2 考查 文化素质选修课 ≥1 2 10420262 数理方程引论 2 2 考查 20240013 离散数学(1) 3 3 考试 3选1 选1门，详见2选1，大学 2选1，一元微

人教版英语必修2课文—原文

2－1 IN SEARCH OF THE AMBER ROOM Frederick William Ⅰ，the King of Prussia , could never have imagined that his greatest gift to the Russian people would have such an amazing history . This gift was the Amber Room , which was given this name because several tons of amber were used to make it . The amber which was selected had a beautiful yellow-brown colour like honey . The design of the room was in the fancy style popular in those days . It was also a treasure decorated with gold and jewels , which took the country's best artists about ten years to make . In fact , the room was not made to be a gift . It was designed for the palace of Frederick Ⅰ. However, the next King of Prussia , Frederick William Ⅰ,to whom the amber room belonged, decided not to keep it. In 1716 he gave it to Peter the Great. In return , the Czar sent him a troop of his best soldiers. So the Amber Room because part of the Czar's winter palace in St Petersburg.About four metres long, the room served as a small reception hall for important visitors . Later,Catherine Ⅱhad the Amber Room moved to a palace outside St Petersburg where she spent her summers. She told her artists to add more details to it .In 1770 the room was completed the way she wanted . Almost six hundred candles lit the room ,and its mirrors and pictures shone like gold. Sadly , although the Amber Room was considered one of the wonders of the world , it is now missing . In September 1941, the Nazi army was near St Petersburg . This was a time when the two countries were at war . Before the Nazis could get to the summer palace , the Russians were able to remove some furniture and small art objects from the Amber Room . However , some of the Nazis secretly stole the room itself . In less than two days 100,000 pieces were put inside twenty-seven woooden boxs . There is no doubt that the boxs were then put on a train for Konigsberg, which was at that time a German city on the Baltic Sea . After that, what happened to the Amber Room remains a mystery . Recently , the Russians and Germans have built a new Amber Room at the summer palace . By studying old photos of the former Amber Room , they have made the new one look like the old one .In 2003 it was ready for the people of St Petersburg when they celebrated the 300th birthday of their city . A FACT OR AN OPINION? What is a fact? Is it something that people believe? No. A fact is anything that can be proved. For example, it can be proved that China has more people than any other country in the world. This is a fact. Then what is an opinion? An opinion is what someone believes is true but has not been proved. So an opinion is not good evidence in a trial. For example, it is an opinion if you say “Cats are better pets than dogs”. It may be tr prove. Some people may not agree with this opinion but they also cannot prove that they are right. In a trial, a judge must decide which eyewitnesses to believe and which not to believe. The judge does not consider what each eyewitness looks like or where that person lives or works. He/she only cares about whether the eyewitness has given true information, which must be facts rather than opinions. This kind of information is called evidence.