Cryptography Quilt – Mathematical Connections

The padlock in the center of the quilt immediately grabs your attention and signals the theme of the quilt: secrecy
and security. The main fabric of the padlock is the same fabric used for the sashing, tying the whole quilt together
visually. All around the padlock are 0s and 1s: a binary code. Can you figure out what it says? As a hint, the binary code uses 8-bit ASCII encoding!

This block illustrates the concept of public key cryptography, and RSA in particular. The shaded items are private information, but everything else in the speech bubbles is publicly stated for anyone to hear. Using the private information, most notably the factorization of n into primes, the chipmunk is able to decode the squirrel’s message. We decided to use the language of flowers for the message since this scene relates to the garden. We made the final design decision in early January, shortly after the riot at the Capitol, and chose the lotus blossom to represent calm and peace.

Tess’s story was the rst scene developed for the Mathemalchemy installation. In the quilt block representing her scene, we have her kite with a key dangling off it. This is a reference to a key that can unlock the central padlock, but also to the Benjamin Franklin kite and key experiment to explore electricity. With this electricity, Tess sparked the development of all the other scenes.

This block shows the “encoding” step of the Hamming code, an error correcting code. The idea behind error correcting codes is that the message may be corrupted in transmission, and the receiver will need to have a way of correcting what they receive to recover the original message. In this block, the person sending the message 0111 adds extra digits that will help the receiver make sure that they can figure out what the message was, even if a digit gets changed accidentally. For the decoding step, see Block K!

This block gives a stylistic representation of blockchain, the mathematics behind cryptocurrrencies such as bitcoin.

Blaise de Vigenère is known as the originator of the polyalphabetic substitution, having written Traicté des Chiffres, in 1585 in which he catalogs cipher methods as well as giving a key to solving polyalphabetic ciphers. In the Vigenère cipher, you use several different shifts of the alphabet to encode your message, making it more complicated than the Caesar cipher, which only uses one shift.

Elliptic curves are deffined by equations of the form y² = x³ + ax + b. Elliptic curve cryptography is a public-key cryptosystem making use of the fact that there is a way to add two points on an elliptic curve to get a third point. The block shows how to add a point P to itself.

Modern cryptography relies on computers and therefore on electricity. This block also relates to the block with Tess’s kite and key.

One way that people kept messages secret in times before modern cryptography was with letter locking|intricate ways of folding and sealing their letters so that it would be clear if someone had opened the letter. This was popular in the 17th and 18th centuries, when letters were folded into themselves without envelopes. Many people developed their own unique ways of locking their letters. Pigeons were used to carry messages between people across long distances from ancient times until telephones were introduced.

In the knapsack problem, one person puts a set of weights into the bag, and the question is whether you can tell which weights were used if all you know is the possible weights used and the total weight of the bag. For example, if you knew that the weights could be 1, 2, 5, or 10 pounds, and the knapsack weighed 16 pounds, could you tell which weights were used? What if each weight can be used at most once? In a simple knapsack cryptosystem, the hidden message is which weights are in the knapsack and the encrypted message is the total weight. While simpler knapsack cryptosystems have been broken, some modern knapsack cryptosystems are believed to be good candidates for post-quantum cryptography.

Lattices-based cryptosystems such as GGH and NTRU use difficult problems about lattices to hide messages. One difficult lattice problem is the shortest vector problem, which in a simple example asks you to find an integer combination of vectors (for example, (4; 7) and (7; 11)) that has smaller coefficients. Sometimes, a “bad” basis (like the vectors given) is the public key that is used to encrypt, while the secret “good” basis of shorter vectors is the private key that is used for decryption. In this block, we can see a lattice with the public basis of longer vectors clearly indicated, while the private basis is less obvious. Lattice-based cryptosystems are important candidates for post-quantum cryptography.

DNA is the code of life! This block also references the Knotical scene
of the installation (with the ship in the bay) with the entanglements of knots. DNA is knotted to fit inside cells, and it is a tricky mathematical problem to figure out how to “unknot” the DNA to replicate it.

While Julius Caesar mentioned the method of delivery of a campaign message in his Gallic Wars, it was Suetonius who related the Caesar wrote to friends by replacing a letter of the alphabet by one 3 places further on, without changing the order of the alphabet.

In this block, we see the receiving end of Block A, the top-left block. The message from Block A (which was padded with extra information) was received with an error, and this block shows the error correction step. After some computation, the original message is recovered. Using error-correcting codes like the Hamming code allows us to ensure that the messages received are the same as the ones intended. They are used in many applications, such as check digits in credit card numbers or bar codes. The message in our two blocks is a reference to the month we finally met to complete the installation, and it is considered a lucky number!

Some cryptosystems rely on the idea that factoring large numbers is a difficult problem. For example in RSA, a public key is a composite numbers which is the product of two large primes. The two primes are the secret private key used to decrypt messages, so if it were possible to factor easily, anyone would have the information they needed to decode a message. Pollard’s rho algorithm is a method for factoring numbers which are the product of two primes. It is named for the Greek letter rho (ρ) because it is an iterative process that eventually creates a cycle, which can be visualized like the shape ρ. And in case you were wondering, this algorithm is not efficient enough to attack RSA given the size of the primes currently being used.

Turning at disk ciphers to rotor disks happened as early as Thomas Edison, though it’s unknown how much use anyone made of such a machine. The most notorious known use is the lag which told the rotors how to shift and rotor system of the Enigma Machine used by both sides in WWII.

The Pigpen cipher is one of those ciphers whose origins are unknown due to its simplicity and of little use in modern society. The base is two tic-tac-toe grids as well as two large X grids with the second of each with a dot in the grid. A letter of the alphabet is assigned to each location, and replicating its “pen” is used to represent the letter.

This block shows the special structure of the integers modulo a prime number (in the case of this particular block, the prime is 19). If you exclude 0, every number modulo 19 is a power of 2; this is what we mean when we say 2 is a primitive root modulo 19. Every prime has a primitive root, and primitive roots are used, for example, in Diffie-Hellman key exchange, where two people can communicate information publicly, but end up with a shared secret.

The ancient Inca used patterns of knots in strings as a way to encode numerical data. As in a historical quipu, the knots in this block record numbers and multiple sets of strands add to the same value. It is thought that this was a kind of bookkeeping or error correction.

This block represents the security of cryptography. We rely on cryptography to protect our secrets, like our credit card numbers and other personal information. Secret stealing seems medieval, and we need a defense.

To obscure the message, it is written on a thin strip of paper while it is wrapped around a stick. Once the paper is removed from the stick, the letters are jumbled. The key to decoding the message is what diameter of stick to use. Because of the technical limitation of the time (1st through 12th centuries CE), getting a uniform cylinder of a specific diameter was difficult.

In this block, two knitter are using Morse code in their knitting to have a conversation. One uses cables to encode the message; the other uses colorwork. Women hid Morse code messages in their knitting using knots in the yarn during WWI and WWII. Famously, Madame Defarge knit names into her scarves in the Charles Dickens novel A Tale of Two Cities. The knitting patterns for the pieces in the quilt are their own kind of code. You have to know the key to understand the instructions.

Our fingerprints are a code unique to each of us. It was observed by Arab merchants as early as 850 that Chinese merchants used fingerprints to authenticate contracts. Sir Henry T Head saw this in India around 1858 but it was not put on a scientific basic for almost 30 years. Part of the problem was how to index the ridges and patterns. Fictional accounts of fingerprints include Mark Twain’s Pudd’nghead Wilson in 1893 and Sir Conan Doyle in a 1903 Sherlock Holmes story set in 1894. Now many of us use our fingerprints as a key to open electronic devices instead of using a traditional password.