Consider a triangle with vertices at x 1 ,y 1x 2 ,y 2and x 3 ,y 3. If the triangle was a right triangle, it would be pretty easy to compute the area of the triangle by finding one-half the product of the base and the height.
However, when the triangle is not a right triangle, there are a couple of other ways that the area can be found. If you know the lengths of the three sides of the triangle, you can use Heron's Formula to find the area of the triangle.
The triangle can be enclosed in a rectangle. The vertices of the triangle will intersect the rectangle in three places, forming three right triangles. These triangles are denoted A, B, and C in the picture. The area of the triangle we desire will be the area of the rectangle minus the areas of the three triangles. The legs of the three triangles can be found by simple subtraction of coordinates and then used to find the area since the area of a triangle is one-half the base times the height.
The area of the triangle in the middle is the difference between the rectangle and the sum of the areas of the three outer triangles. It turns out that the area of a triangle can also be found using determinants. The derivation of the formula is kind of long and most of you don't care to see it, so it's on a separate page.
It is possible that you will get a negative determinant, like we did here. Don't worry about that. The sign is determined by the order you put the points in and can be easily changed just by switching two rows of the determinant.
Area, on the other hand, can't be negative, so if you get a negative, just drop the sign and make it positive. Finally, divide it by 2 to find the area. Do not say the area is both positive and negative. Why not use absolute value, you ask? Well, think how confusing it would be to have the absolute value of a determinant. Continue with the idea of finding the area of a triangle. If the area of a triangle was equal to zero, then there would be no triangle, the points would be collinear on a line.
Three points are collinear if and only if the determinant found by placing the x-coordinates in the first column, the y-coordinates in the second column, and one's in the third column is equal to zero. You are not setting the determinant equal to zero, you are testing to see if the determinant is zero. Notice this time, that the actual variables x and y are in the determinant.
That is because you have two points on a line given, and the point x,y is a generic point on the line. When you expand this, I strongly recommend that you expand along the first row. That way, your multiplications to find the determinants won't involve x or y.
The derivation of Cramer's Rule can be found on another page. Here are the results. Let D x be the determinant of the coefficient matrix where the x column has been replaced by the constants from the right hand side. Let D y be the determinant of the coefficient matrix where the y column has been replaced by the constants from the right hand side.
Cramer's rule is named after Gabriel Cramer who lived from - To browse Academia. Skip to main content. Log In Sign Up. Prabhavathi S. PrabhavathiM.
Applications of Cryptography includes Electronic Commerce, chip based payment cards, digital curriencies,computer passwords and military communications. Modern cryptography is heavily based on Mathematical Theory and Computer Science practice. One discipline that is sometimes used in Cryptography is Linear Algebra. One method of encryption by using Linear Algebra, specifically Matrix operations.
The method involves two matrices: one to encode, the encoding matrix and one to decode, the decoding matrix. This paper attempts how to derive applications of matrices in Cryptography in day to day life.
Key Words: encoding, confidential, secret coding. Related Papers. By Dr. R Joshi. By Nidhal El Abbadi. By dayyan zahid khan. In Memoriam: Michael Miki Neumann — By Michael Tsatsomeros. By Peter Butkovic. Download file. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link.
Need an account? Click here to sign up.Cryptography is the discipline of encoding and decoding messages. It has been employed in various forms for thousands of years, and, whether or not you know it, is used frequently in our daily lives.
Encryption is used to keep our data safe on the Internet, when we use the ATM, and in many other everyday activities. There are two main categories of cryptographic procedures. A code works by replacing whole words or phrases with others, at the level of meaning; for example, when a parent substitutes one word for another in front of his or her child.
A cipher works by transforming and replacing individual letters. Before we go deeper into the details of cryptography, there is some associated terminology that you should be familiar with. The alphabet is the characters that the message will be written in. Plaintext or clear text is the original message in readable form.
We will refer to encoding the message as encryption or enciphering. The key or password is the information that is used in the encryption process and is the basis of security for a code. These keys are usually kept private, and the inverse key used to decipher the encoded message is given to the person needing to decode the message.
The encrypted and hard-to-read message is called ciphertext or an encrypted message. Using the inverse key to turn the ciphertext back into plaintext is referred to as decrypting or deciphering. If you are able to decrypt the message without being told the inverse key, we call that cracking the code.
A Hill cipher is a type of polygraphic cipher, where plaintext is divided into groups of letters of a fixed size and then each group is transformed into a different group of letters. A Hill cipher accomplishes this transformation by using matrix multiplication. It was one of the first practical applications of linear algebra to polygraphic ciphers. Hill ciphers were first described by their creator Lester Hill in in The American Mathematical Monthlyand he wrote another article about them in Hill ciphers are applications of linear algebra because a Hill cipher is simply a linear transformation represented by a matrix with respect to the standard basis.
Groups of letters are represented by vectors. The domain of the linear transformation is all plaintext vectors, while the codomain is made up of all ciphertext vectors. Matrix multiplication is involved in the encoding and decoding process. And when trying to find the inverse key, we will use elementary row operations to row reduce the key matrix in order to find its inverse in the standard manner.To browse Academia. Skip to main content. Log In Sign Up.
Sadiq Shehu. Also the area of cryptography employs many different means of transforming normal data in to unreadable form. This paper describes an activity build around one of the techniques that illustrates an application of matrices to cryptography. The method involves two matrices of which one is used to encode the encoding matrix and the other one to decode the decoding matrix. The characters, in the original message or stream are assigned numerical values and the matrix must be invertible for use in decoding.
The proposed method is very simple in its principle and has great potential to be Applied to other situations where the exchange of message is done confidentially. It allows people to do business electronically without worries of deceit and deception in the distant past, cryptography was used to assure secrecy. When people started doing business online and needed to transfer found electronically, the application of cryptography for integrity began to surpass it use for secrecy.
Hundreds of thousands of people interact electronically every day, whether it is through e-mail, e-commerce business conducted over the internetATM machines or cellular phones. The constant increase of information transmitted electronically has lead to an increased reliance on cryptography and authentication. Cryptography mainly consists of encryption and decryption. Encryption is the transformation of data in to some unreadable form its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended even those who can see the encrypted data.
Encryption and decryption require the use of some secret information, usually referred to as a key depending on the encryption mechanism used, the same key might be used for both encryption and decryption, while for other mechanisms, the keys used for encryption and decryption might be different.
Hence the proof. Convert the text message of length L in to a stream of numerals using a user friendly scheme for both the sender and receiver. Convert the text message of length L in to a stream of numbers that contains the encrypted message and send to the receiver. Decoding process 1. Place the encrypted stream of numbers that represent the encrypted message in to a matrix.
Multiply the encoded matrix X with the decoder A-1 the inverse of A to go back the message matrix M 3. Convert this message matrix in to a stream of numbers with the help of the originally used scheme 4. Also we assign the number 0 to a blank or space between two words. Here the security is assumed as only those know about the quadratic form can understand the process.
Cryptography Using Matrices in Real Life
Gnanaraj; international journal of computer technology. S, Vasta Suchi. Related Papers. By eSAT Journals. Download file. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up.Perhaps when you were a child, you had a decoder ring from a cereal box or Cracker Jacks. If you wanted to send someone a secret message, you could use the ring to encode your message and send the encrypted message to your friend.
Your friend uses the same ring to decode the message. In the movie "A Christmas Story", the hero sends in Ovaltine labels to get a decoder ring so he can decode the secret messages from Orphan Annie which were broadcast on the radio show. The basic idea of cryptography is that information can be encoded using an encryption scheme and decoded by anyone who knows the scheme. There are lots of encryption schemes ranging from very simple to very complex.
Most of them are mathematical in nature. It's not difficult to guess at the importance of this application. It is surmised that Hitler may well have won WWII had not a bright British mathematician been able to break the codes used by the Axis forces in their radio transmissions. Today, sensitive information is sent over the Internet every second: credit card numbers, personal information, bank account numbers, letters of credit, passwords for important databases, etc.
Often, that information is encoded or encrypted. We are going to play with matrix encryption. The encoder is a matrix and the decoder is it's inverse. Let's see just how that works. Let A be the encoding matrix, M the message matrix, and X will be the encrypted matrix.
Then, mathematically, the operation is. Note, the sizes of A and M will have to be consistent and will determine the size of X. OK, let's say someone has X and knows A. That would be the same as solving the matrix equation for M. Multiplying both sides of the equation on the left by we have. Notice then that the encoding matrix, A, must have an inverse for this scheme to work. Let's do an example.
The first thing to notice is that letters aren't numbers and so we'll have to assign them. Make a list, you will need it for the assignment. We are going to need blanks, too, so let a blank be represented by 0. Does anyone know what event made this line famous? We need to translate letters into numbers.
Using the list above, the message becomes:.RSA Matrix Encryption Video Presentation - CSCI 4315
Now we need to decide on a coding matrix. I'm going to suggest. Since this is a 4 x 4 matrix, we can encode only 4 numbers at a time. We break the message into packets of 4 numbers each, adding blanks to the end if necessary.
The first group is 20, 8, 5, 0. The message matrix will be 4 x Cryptography, to most people, is concerned with keeping communications private. Indeed, the protection of sensitive communications has been the emphasis of.
Encryption is the transformation of data into some unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data. Decryption is the reverse of encryption; it is the transformation of encrypted data back into some intelligible form. Encryption and decryption require the use of some secret information, usually referred to as a key.
Depending on the encryption mechanism used, the same key might be used for both encryption and decryption, while for other mechanisms, the keys used for encryption and decryption might be different. We assign a number for each letter of the alphabet. For simplicity, let us associate each letter with its position in the alphabet: A is 1, B is 2, and so on.
Also, we assign the number 27 remember we have only 26 letters in the alphabet to a space between two words. Thus the message becomes:. Since we are using a 3 by 3 matrix, we break the enumerated message above into a sequence of 3 by 1 vectors:. Note that it was necessary to add a space at the end of the message to complete the last vector. We now encode the message by multiplying each of the above vectors by the encoding matrix.
This can be done by writing the above vectors as columns of a matrix and perform the matrix multiplication of that matrix with the encoding matrix as follows:. The columns of this matrix give the encoded message. The message is transmitted in the following linear form. To decode the message, the receiver writes this string as a sequence of 3 by 1 column matrices and repeats the technique using the inverse of the encoding matrix.
The inverse of this encoding matrix, the decoding matrix, is:. Thus, to decode the message, perform the matrix multiplication. Cryptography - An Overview. Elementary Cryptanalysis a book on Cryptography.Cryptography is concerned with keeping communications private.
6.5 - Applications of Matrices and Determinants
Today governments use sophisticated methods of coding and decoding messages. One type of code, which is extremely difficult to break, makes use of a large matrix to encode a message. The receiver of the message decodes it using the inverse of the matrix. This first matrix is called the encoding matrix and its inverse is called the decoding matrix.
To encode a message, choose a 2x2 matrix A that has an inverse and multiply A on the left to each of the matrices.
If you dont know the matrix used, decoding would be very difficult. When a larger matrix is used, decoding is even more difficult. But for an authorized receiver who knows the matrix A, decoding is simple. The receiver only needs to multiply the matrices by A-1 on the left to obtain the sequence of numbers.
The message will be retrieved with reference to the table of letters. Polygraphic systems encode a group of plain sequence letters. This scrambles the frequencies and allows for more than one representation of a plain sequence character. The digraphic system is the simplest polygraphic system. It uses a 2 x 2 coding matrix to replace pairs of plain sequence characters.
A square matrix of any size may be chosen as a coding matrix.
The larger the coding matrix the more complex the system of cryptography. The examples in this unit use a trigraphic system. A 3 x 3 matrix is chosen as the coding matrix. The choice of the matrix is arbitrary.
The only constraint is that the coding matrix must have an inverse. Therefore, we have added a space and two punctuation marks to the standard alphabet to create our plain sequence. The plain sequence and its numerical representations are. For the message, Imagination is mo, we will encode groups of three letters.
This system is called trigraphing. Be sure the matrix is 3x3 and has an inverse. Learn more about Scribd Membership Home.