Module 2 Summary: Module 2 Summary

1. Module 2 Summary

Unit 2 Conclusion

Module 2 Summary

Entering a password before being admitted to a website is a normal practice in today’s security-sensitive world. The development of codes that are extremely difficult to decipher is helping to make this practice a safe one. It is possible to determine the key for a cipher by the method known as brute force, where every possible key is tried. However, the use of 128-bit encryption, where there are 2128 possible combinations, has been regarded as a highly effective way to deter would-be criminals who would use such a method.

Prime numbers are a key part of modern encryption methods. The product of two very large prime numbers can be used as a key to encipher and decipher a secret message. The benefit of this technique is that it is very easy for the makers of the code to record or simply recall the two prime numbers that are used. At the same time, it is very difficult for an intruder to obtain the prime factorization of a product of two large primes. This is then an easy code to encipher but a difficult one to decipher without knowledge of the key.

Mathematics and cryptography share a common characteristic. Both disciplines make use of inverse processes. In cryptography you need to be able to both encipher and decipher a message. The steps for deciphering are a reversal of the steps for enciphering. Similarly, in mathematics, whenever a number undergoes a particular operation, the result can be reversed and the original number recovered. This commonality between the two disciplines makes mathematics an ideal basis for cryptography.

In this module you learned how to obtain the prime factorization of composite numbers. You learned how to use prime numbers to develop simple ciphers. You also used various strategies to identify perfect squares and perfect cubes and to evaluate their roots. You examined irrational numbers and learned how to locate them on a number line by various means. You also learned that the properties of irrational numbers are well suited as keys for secret codes.

In the second half of the module you investigated radicals. You saw how mathematical processes are reversed by learning how to convert entire radicals into mixed radicals and vice versa. You reviewed the exponent laws and discovered how you can interpret negative, zero, and rational exponents. Later, you learned how to convert powers with rational exponents into equivalent radicals that enabled you to evaluate the expression more easily. You also worked with your peers to detect errors in your work and helped each other to avoid committing similar mistakes in the future.

Go to Module 2: Learning Outcomes. The learning outcomes for this module are summarized in the chart. As a review of what you have learned, complete the chart by identifying those activities you undertook to address the corresponding outcomes. An example of what the chart could look like can be found in the Module 1 Summary. Please save a copy of this completed chart with your work from this module in your course folder.