Alexander Zhang

When you open your computer and type in your password, you are using cryptography. More specifically, you are decrypting a code to authenticate that you are the rightful user of this computer. Cryptography is the science of writing, using, and solving codes. Humans have been using codes for a very long time, starting from 1900 BC when an Egyptian scribe used non-standard hieroglyphs in an inscription. Inscriptions are words inscribed or written in a book or monument.
The five main functions of cryptography are privacy, authentication, integrity, non-repudiation, and key exchange. Privacy is for ensuring that no one can has access to the message except for the intended receiver. Emails utilize this privacy, so only your intended receiver can see your message. Authentication is used to prove one’s identity, so only the right identities can access something. Locks, passwords, and key codes all examples of authentication devices. Integrity assures that the receiver of a message can be sure that the original message had not been altered in any way. Without integrity, we wouldn’t know whether or not to trust emails we receive because they may have been altered. Non-repudiation is a mechanism that proves that the sender really sent this message. If you receive a message, you may think a certain person sent it. However, the reality is that any individual may have deceived you in creating a false email. Non-repudiation prevents this from happening. Key exchanges allow the sender and receiver to communicate crypto keys, allowing the receiver to decipher the message.
In cryptography, we start with a piece of understandable text, referred to as plaintext. Plaintext is encrypted into ciphertext using a key. Ciphertext is just the result of plaintext being encrypted. The encrypted message is not understandable without a key, which is used to decrypt the message back into readable and usable plaintext.
The encryption formula is: ciphertext = encryptionkey (plaintext). The decryption formula is: plaintext = decryption methodkey (ciphertext). The key is just an algorithm applied to the plaintext or ciphertext. So to get ciphertext, you just apply an algorithm on some plaintext.
Another encryption method is to multiply two very large prime numbers. It is very easy to multiply them, but it would be very hard to find the two original prime numbers. Take 4320 for example. Try to list a couple of factors, it’s very easy. 1, 2, 3, 4, 5 etc… There are many factors in this number: 60 and 72 are factors of 4320, and they each have many factors. To find all the factors of 4320, most people would prime factorize and draw a factor tree. But that won’t be easy if the only factors of a number are two big prime numbers. Try finding all the factors of 4453. 4453 is not much greater than 4320, but it is much harder to list its prime factors. A factor tree would definitely not help. The prime factors of 4453 are 61 and 73. There is no easy trick to factor numbers, so it can take a computer days or even years to use trial and error to factor a HUGE number. Imagine trying to find the two prime factors of 144,483,604,528,043,653,279,487! That number is the product of a 10 digit prime number and a 14 digit prime number. The product can be the public key, and the factors can be communicated as private keys between sender and receiver. As you can see, math is very useful in cryptography, especially large prime numbers. Prime number cryptography is very prominent. The RSA cryptosystem uses this method. It is widely used for secure data transmission.
Cryptography is very important. Without it, we would not be able to trust and believe anything we see on the internet. Examples of cryptography include ATM cards, computer passwords, and electronic commerce. Simple early forms of cryptography had people replacing a letter by the letter that precedes or follows it in the alphabet. In fact, Julius Caesar shifted letters three times when communicating with his generals.
Cryptography was used in World War II a lot, where both Allied nations and Germany were sending encrypted messages. Cryptography played an important role in the allies win. The German’s "Enigma" Cipher was successfully decrypted by the Alan Turing and the Allies. The decryption of Enigma allowed the allies to read important parts of German radio traffic, so the allies knew the Germans’ plans. It was an invaluable source of military intelligence throughout the war. The US Army Signals Intelligence Service also broke into PURPLE, an important Japanese cipher. The advanced deciphering technology of the Allies helped them win the war.
Steganography is the practice of concealing messages within non-secret texts. It was developed in ancient times, where messages were tattooed on skin and later hidden by hair. Modern examples of steganography include the use of invisible ink, microdots, and digital watermarks. A microdot is a microphotograph, especially of a printed or written document, that is only about one millimeter across. A digital watermark is a kind of marker that is embedded in a noise-tolerant signal such as an audio, video or image data. Hidden messages were written with invisible ink between lines of common text. The advantage of steganography over cryptography is that secret messages and information did not attract attention.
The two biggest types of cryptography are symmetric-key cryptography and public-key cryptography. In symmetric-key cryptography, both the sender and receiver share the same key to encrypt and decrypt a message. A cipher is a pair of algorithms that create the encryption and the reversing decryption. There are two types symmetric key ciphers, block ciphers and stream ciphers. A block cipher enciphers inputs as blocks of plaintext. A stream cipher enciphers input as individual characters. In public-key cryptography, also called asymmetric-key cryptography, two different keys are used. The two mathematically related keys are the public key and the private key. The public key can be freely distributed and is used to encrypt the message, while the private key must remain secret and is used to decrypt the message.
We use many of cryptography's five main functions in our daily lives, especially authentication and privacy. These codes have been utilized since our early days, yet we have advanced these aspects of our lives to make it far more advanced.


Products of large primes,
Very hard to factor fast.
Primes in encryption.