Computers and Mathematics

Transcription

1 Computers and Mathematics Benjamin Walters Bauer Team 1B Abstract Computers and Mathematics have been deeply intertwined since the invention of computing. The first computers were invented for the sole purpose of solving complex mathematical equations. Since then, computers have been used to solve age-old previously unsolvable or unprovable mathematical problems. Mathematics remains a large component of computer science today. Current topics in mathematics include theoretical computer science, scientific and High Performance Computing, and cloud computing and scalability algorithms. Another important topic in computers and mathematics is cryptography. This paper discusses these topics in depth and proposes solutions for making progress in these fields. Index Terms Computers, Mathematics, Algorithms BACKGROUND Since the dawn of computers, mathematics has been the most fundamental component of the field. During World War II, the U.S. military employed many calculators. These calculators were skilled employees with desk calculators tasked with computing missile trajectories to be used in battle. A 60 second trajectory could be calculated by a human in about 20 hours. A special purpose machine, the Bush analyzer, was created to speedup these calculations. This machine could calculate the same trajectory in 15 minutes. While this was a significant improvement that didn t require human workers, the military wanted to do better. [1] In an effort to improve this calculation time, the Electronic Numerical Integrator And Computer (ENIAC) was developed. Although it was developed to calculate firing tables, it was not finished until after World War II. Although it was not able to be used in time, ENIAC could calculate a 60 second trajectory in only 30 seconds, less than the flight time. This is 2400 times faster than the human solution and 30 times faster than the previous computer solution. However, the calculation speed is not the most impressive thing about ENIAC. Unlike the Bush analyzer, ENIAC was not built specifically to calculate firing tables. It could perform a wide array of numerical calculations. Thus, ENIAC became the world s first general-purpose electronic computer. [1] SOLVED PROBLEMS Computers helped not only with pure numerical calculations but also with more theoretical mathematical problems. For example, an extremely famous problem in mathematics is the four color theorem. It was proposed as early as 1840, far before ENIAC was developed. The theorem states that any map can be colored with as few as 4 colors such that each no adjacent territories are the same color. However, it remained unsolved until 1976, when the theorem was proved with the aid of computers by Kenneth Appel and Wolfgang Haken at the University of Illinois. The theoretical portion of the proof showed that each component of a map could be in one of around 1900 (later reduced to 1400) configurations. They then wrote a computer program to prove that each of the configurations could be colored with only 4 colors. [2] Appel and Haken s computer aided proof was met with intense criticism from the mathematics community. People were very skeptical of the validity of a computer produced mathematical proof. The proof was so difficult to believe largely because it was not very intuitive. This forces mathematicians to trust that the program was constructed correctly and that nowhere does it have any incorrect logic. However, in 2005, Benjamin Werner and Georges Gonthier developed another proof of the four color theorem using the Coq proof assistant. This reduced some of the skepticism over the proof because it only required trust in the validity of the Coq system. This is easier to trust because it has a smaller set of well formulated logic rules. These types of logical automated proof systems have been more accepted, so this proof was more accepted. [2] Another area of mathematics greatly affected by the invention of computers was cryptography. Cryptography is mainly concerned with encryption, or the encoding and decoding of secret messages. Evidence of encrypted messages goes as far back as 1900 BC [3]. Encryption allows one person to send a secret message to another person. The message cannot be read by anybody except the intended recipient. To do this, the sender encodes the message. Once encoded, the message will appear to be gibberish and will not represent the original message whatsoever. To read the message, the recipient must decrypt the message. This will restore the original message. Throughout history, there have been many different encryption techniques. Many of the techniques, such as the Caesar cypher, relied on the fact that only the sender and receiver knew the technique to encode and decode the message. This is not entirely reliable since anyone who figures out the method can read the message. While some more CS XXX 1

2 complex and reliable techniques did arise, it wasn t until the invention of computers that cryptography became accessible to so many people. Computers provide so many more methods of message encryption. One of the most common modern encryption cyphers is Advanced Encryption Standard (AES). One modern method is called symmetric-key cryptography [3]. This method uses a shared key to encrypt the message. Once the message is encrypted, it can only be decoded by that same key. The difficult with this method is that a sender and receiver have to have already reliably and secretly shared this key with each other beforehand. Another method that avoids this problem is called public-key cryptography [3]. This method involves a key pair. The sender uses the receiver s public key to encode the message. Then, the encrypted message can only be decoded by the receiver s private key. This is an ingenious method. The receiver s public key can be posted online so anybody can encode a message for them. However, only they can decrypt these message with their key. The strongest part of both of these methods is that the method is not secret. This allows us to develop open-source code for these types of encryption. Thus, encryption is now provided to nearly anyone with access to a computer. CURRENT PROBLEMS Though computers have helped us to solve many problem in mathematics, there are still many fields in which we are only in the early days. Most problems solved by computers give us the ability to explore even deeper and more important problems. Some major problems that currently exist are theoretical computer science, scientific and high performance computing, and cryptography. While computers have had a major impact on traditional mathematics, they have spawned a new hybrid field of mathematics and computer science called theoretical computer science. This field has to do with algorithm design and computational complexity. There are still many open problems in this field [4]. One classic problem is the traveling salesman problem [5]. The problem has a list of cities and the distances between them. The goal is to find the shortest route to visit each city once. The seemingly simple is what is known as NP-complete, which means it cannot be solved in polynomial time. These types of problems cannot be run on very much data because they would take hundreds of thousands of years on even our fastest supercomputers. Although this seems like a problem with a small scope, it can actually be applied to many similar problems such as logistics, circuit design, and networking. In the early days of computing, programs only ran a sequential set of instructions. Each program would just be a list of instructions to be run in order. This change with the invention of the multi-core processor. With multiple hardware threads, processors could now execute multiple instructions simultaneously. Over the years, the number of cores per processors and the number of processors per computer have been growing. On consumer hardware, we now can get upwards of 4 processors and more than 8 cores per processor. However, if we connect many computers together, we can develop applications that leverage the processing power of many computers. This field is known as High Performance Computing. High Performance Computing allows us to carry out computations in hours that would take years on single machines. Tianhe-2, the largest supercomputer in the world, has over 3 million CPU cores and can perform 33 * operation per second [6]. The biggest field that utilizes supercomputing is scientific computing. This largely consists of scientific simulations on large amounts of data. Many scientific simulations require extremely large amounts of computation on similarly large amounts of data. While we are consistently able to increase the number of cores in a supercomputer every few years, we are struggling to make software that can efficiently utilize all of the cores. Although much work has been done to create scalable algorithms, there are certainly many open problems in writing applications that can scale to millions of cores. Earlier we discussed the vast progress that has been made in cryptography since the invention of computers, but now we will discuss the problems and weaknesses with cryptography. Both techniques discussed earlier, symmetric-key and public-key cryptography, rely on the computational complexity of factorization of extremely large numbers [7]. The keys in cryptography are constructed out of numbers that are extremely difficult to factor. However, if an efficient method were developed for factoring these numbers, these types of cryptography would become ineffective. Therefore, more methods of cryptography will be needed to maintain reliable encryption. EFFECT ON SOCIETY Although computers as we know them today have been around for under 100 years, they have already drastically changed our society. For most, they have become an integral part of daily life. They have truly transformed how we learn, interact, relax, and do business. We will discuss several major changes and their upsides and downsides for society. Arguably one of the most important ways that computers have changed society is how they have revolutionized the way we do business. Many day-to-day business management tasks used to be done by hand, such as bookkeeping, payroll management, and employee scheduling. These required people, often with simple desk calculators, to perform these tasks by hand. Now, nearly every business has some way to CS XXX 2

3 computerize these tasks. This has both a positive and negative effect. As a positive for the business, this reduces the labor cost for managing these tasks. Before computers, large businesses would need to hire many people to do these tasks. However, with computer software, businesses can have a single person or a small group manage all of these tasks via software. Another positive is the reduced likelihood of mistakes. Instead of a human doing many calculations by hand and transferring different numbers between different forms and sheets, a computer is managing these tasks. While you still have to trust the programming of the software, software at least is has a reproducible and testable effect. However, there are of course also negative effects on business. By reducing the number of employees needed to manage these sorts of tasks, this software reduces the number of jobs, putting some employees out of work. Many will argue, however, that it also creates jobs. Someone still needs to program the software. Someone still needs to use the software at the business. However, by leveraging the work of the computer, much of the work can be done by many fewer people. Therefore, this software results in an overall net loss of this type of job. In addition, although there are fewer employees required, employees must be retrained. For example, the software developer needs to be educated in programming. The user of the software also needs to be trained on using the software. While this job loss seems bad, it may allow businesses to redirect the money that would be going to those employees salaries into other areas of the business. For example, the business could spend more on research and development as opposed to those simple management tasks. This software allows businesses to spend less money on busy-work and more money on tasks directly related to the service they work to provide. In summary, mathematics software in business often causes low-skill jobs to be replaced with more specialized jobs. With the rise in popularity and decrease in price of computers, computers are no longer a product only affordable by business. Computers have become integral parts of the everyday life of many people. In 2013, according to the U.S. Census Bureau, 83.8% of households in the U.S. had a computer. In addition, 74.4% of total households had internet access [8]. Computers, and the internet especially, have given many people access to information at their fingertips that was never available before. It also allows people from all over the world to be far more connected than ever before possible. People can easily keep in touch with friends and family from all over the country and world like never before. This has created an increasingly connected world. However, with all these benefits come some definite downsides. While people are able to connect to those far away from them, they can sometimes become too engaged in their devices and less engaged in what is happening around them. People will often stay engaged with their devices while they would normally be socializing with others. For example, a group of friends out to dinner together will often have several friends checking their phones every few minutes. While this is not directly stemming from mathematics, advancements in mathematics are the reason that such great interconnection through the internet is possible. Next, we will discuss more direct effects of mathematics. Cryptography is another vastly changing field that has had an extreme effect on society. Cryptography has allowed for intense innovation, especially in business. The ability for businesses to send encrypted messages has allowed national and international commerce to flourish. Large companies with many branches can electronically communicate confidential business details without worry of interception since the data is encrypted. In addition to general purpose businesses, cryptography has massively transformed the stock and banking industry. A large portion of banking is all done electronically now. All stocks are traded electronically. Without encryption, none of this would be possible. Banking data can be safely and secretly transmitted across the country in an instant. A stock on the New York Stock Exchange can be bought and sold in seconds from anywhere in the country. This has not only transformed business, but individual consumer life as well. Any bank customers with computer and internet access can check and manage their balance from a computer or smartphone. While this provides great convenience, it has also created a major negative effect. The engagement of technology in everyday business and personal life has created a vast over-reliance on technology. When nearly every step of a business s operations rely on technology, if that technology ever fails, the business is hurt by it. For example, when the internet goes down at one branch of a company, it communicate as easily with both customers and other branches of the company. If servers go down, customers can lose access to that business s services. In terms of personal life, if somebody s computer or phone breaks, they may lose access to many important things. They may not be able to communicate with children, parents, or spouses. They may not be able to access for work. Many things rely on technology in ways that they haven t in the past. This is one reason that cryptography is so important. Without reliable encryption, much of the business and financial infrastructure in the U.S. would come to a screeching halt. That is why the possible issues with cryptography discussed previously are so dangerous. Advancements in computers and mathematics have vastly changed our society in the relatively short time that computers have existed. While they have improved many lives and added many conveniences that were not possible before, they have created an over-reliance on that technology. However, there doesn t seem to be a way to move forward without the aid of CS XXX 3

4 technology, so we will just have to try to find other ways to reduce our dependence on technology. FUTURE PROBLEMS AND PROPOSED SOLUTIONS While we have talked about the current open problems in computers and mathematics, the problems are changing very rapidly. While many of the classical unsolved problems remain, the majority of problems that arise will get solved and lead to new problems. Therefore, most of the problems that exist in the field will likely be very different ten years from now. In this section, we predict what will be some of largest problem areas moving forward and proposed solutions for these problems. With decreasing cost of small computing devices, we are seeing the rise of the Internet of Things (IoT). The idea of IoT is that nearly every device we use in the future will be in some way connected to the internet or at least other devices around it. For example, your oven will be able to text you when your food is finished cooking. Or your fridge will automatically order from the grocery store when you finish the last soda. This will demand very flexible and scalable networks. We may have to evolve our concept of a network. Maybe the old model of Internet Protocol (IP) and Local Area Networks (LAN) will not be the most efficient mechanism for these many-device small networks that will be connecting to the internet. I think there will be major innovation in the network space over the next years, as it will continue to be the factor that drives personal device innovation. While many technologies fall by the wayside all the time, one technology that is not going anywhere is the cloud. The idea of the cloud is being able to use your own device to connect to many other devices that provide a variety of services, such as storage. This convenience is far too great to go anywhere anytime soon. It allows users to use a fairly weak device, such as a phone, to access seemingly infinite resources provided by the cloud. However, it introduces several major issues. One issue with the cloud, as mentioned earlier, is the dependence it creates. If many services are provided by resources not directly in control of the user, the user has to hope that they will be available when they are needed. To accommodate this, we need to continue to research many fault tolerance methods. Fault tolerance includes policies and mechanisms for providing services when resources fail. This will help to be continually providing services to users even when hardware or software fails. Another group of issues facing the cloud are privacy and security. There have been countless issues with security breaches in cloud services. Since they do not control any of the resources, users are putting a lot of faith into the security of the service providers. While security is simply an issue of the technology being used for the services, privacy is a more complicated topic. While a service may be completely secure, it may not provide privacy in the same way. For example, a cloud storage service may have very advanced security measures, but its service agreement may allow it view the contents of your storage to serve you targeted advertisements. The largest problem moving forward with privacy will be educating users on what privacy they are loses to specific services. However, many people are willing to give up certain privacy in order to use services at a reduced cost. This will certainly be and ongoing battle in the coming years. Another important issue in the future will be cryptography. With the extreme reliance we have on it as a society, it will be very important that we continue to research and make advancements in the field. One other field that may be a threat to it is quantum computing. Quantum computers work very differently than traditional computers. Instead of performing traditional mathematical operations, they exploit the physical characteristics of quantum bits to arrive at the solution to a problem [9]. By approaching problems in an extremely different way, quantum computers have the potential to make major breakthroughs in many fields of computer science. For example, quantum computers would be ideal for solving problems like the traveling salesman problem. However, they also have the potential to solve problems like factorization of large numbers. As mentioned earlier, this would be extremely detrimental to the field of cryptography. However, since quantum computers have the potential to make great strides in other fields of computer science, we should not hold back just to protect modern cryptography. Instead, we should continue to search for alternative methods of cryptography that do not rely on these properties that may be broken by quantum computing. As we can see, computers and mathematics will likely never run out of problems to tackle. Due to the mass adoption of computers, whenever a problem is solved, it will likely lead to new problem areas to explore. Therefore, we will keep up this ongoing cycle of research and development. REFERENCES [1] ENIAC: The Army-Sponsored Revolution, ENIAC: The Army-Sponsored Revolution. [Online]. Available at: [Accessed: 9-Apr-2016]. [2] Four color theorem, Wikipedia. [Online]. Available at: [Accessed: 9-Apr-2016] [3] Cryptography, Wikipedia. [Online]. Available at: [Accessed: 09-Apr-2016]. [4] List of unsolved problems in computer science, Wikipedia. [Online]. Available at: CS XXX 4

5 in_computer_science. [5] Travelling salesman problem, Wikipedia. [Online]. Available at: m. [Accessed: 09-Apr-2016] [6] TOP500ListsNovember 2015, November [Online]. Available at: [Accessed: 09-Apr- 2016]. [7] Integer factorization, Wikipedia. [Online]. Available at: [8] Computer and Internet Use, Main. [Online]. Available at: [9] Quantum computing, Wikipedia. [Online]. Available at: CS XXX 5