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Cryptography

by gadgetic on Mar 19, 2010

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Quick Mathematics Presentation on the basics of Cryptography.

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Quick Mathematics Presentation on the basics of Cryptography.

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Modern cryptography intersects the disciplines of mathematics, computer science and engineering. Applications of cryptography include ATM Cards, computer passwords and electronic commerce. Throughout history Cryptography has evolved from very simple and basic techniques of hiding information to more robust and secure methods using modern technology. Julius Caesar in 44 BC used simple letter substitution to communicate with his generals. Each letter in his message was offset by three, so for example using today's English alphabet the word “help” would be encrypted as “khos”.During the World War 1, breaking enemy codes was at the heart of battle. Many code books were being left behind from sunken submarines or blown up aeroplanes. By World War 2, code books on submarines were bound in lead to help them sink to the bottom of the ocean in the event of an enemy takeover.The invention of computers in the 20th Century revolutionised Cryptography. IBM created a code, the Data Encryption Standard, considered to be unbreakable at the time was trusted by the US for the encryption of unclassified informationit was eventually broken in the late 1990’s and replaced recently by a public competition by the Advanced Encryption Standard.
There are two different types of cryptosystems. Symmetric and Asymmetric....There are some key disadvantages of using the symmetric system which was the method used in traditional cryptography, as in this system both the sender and receiver need to know and use the same key. This presents a challenge of agreeing on a key whilst maintaining secrecy. Plus the key also needs to be shared with both parties, if they are in different locations, then an outside party such as a courier, a telephone system or email must be trusted to send the key. This could easily be intercepted, and the interceptor can read modify and falsify and messaged exchanged between the two parties.Another major shortfall of the symmetric key is that many keys must be shared between parties and the need for key management arises. But how do you manage to store secret keys securely in a trusted manner?!!Hence the introduction of the Asymmetric cryptosystem. Which solves the need for sharing keys and key management.
The public key is published, while the private key is kept secret. The need for the sender and receiver to share secret information is eliminated; all communications involve only public keys, and no private key is ever transmitted or shared. In this system, it is no longer necessary to trust the security of some means of communications. The only requirement is that public keys be associated with their users in a trusted (authenticated) manner (for instance, in a trusted directory). Anyone can send a confidential message by just using public information, but the message can only be decrypted with a private key, which is in the sole possession of the intended recipient.The main disadvantage of this system is speed. Large amounts of data will take time to encrypt, which is why the public key system is mostly used for emails. Another disadvantage of a basic public system, is that because everybody knows the receivers public key, they can pretend to be a fake sender. Digital Signature Authentication provides a solution to this problem, this is where the sender decrypts the message first before sending it using their own private key and then the receiver using the senders public key can recode it.
Computers use numbers to represent letters, which means that we need to encode a set of numbers in such a way that using a public formula so as to make it very difficult and near impossible to ascertain the decrypting formula. A system was developed Rivest, Shamir and Adleman abbreviated as RSA.For public key encryption we use the formula e=m to the k mod (n). Where “e” is the encrypted number and “m” the original number.Then for decrypting the number we simply pass “e” into the formula to get m=e to the d mod (n)What makes this encryption hard to break is the fact that it is nearly impossible to derive d by simply knowing n and k. And it is the “d” component which makes up the private key.

Cryptography — Presentation Transcript

Cryptography
From Greek kryptos, "hidden, secret"; and gráphō, "I write", is the practice and study of hiding information.
44 BC – Julius Caesar – Letter Substitution
“H E L P” encrypted as “K H O S”
World War Code Books
Personal Computers
IBM - DES – Data Encryption Standard 1976
Public Competition- AES – AdvancedEncryption Standard 2002
Gurpreet Sidhu
Cryptosystems
Symmetric
Use the same key to both encrypt and decrypt a message.
- Sender and Receiver both need to agree on the same key
- Must trust some courier, phone line or email to send key
- Prone to interception and hence message is no longer secure
- Many keys needed between parties and stored in a trusted manner!
Asymmetric
Use a public key to encrypt and a private key to decrypt messages.
Public Key Cryptography
Whitfield Diffie and Martin Hellman introduced the concept of public-key cryptography in 1976
Hello
Public Key
Receiver
Sender
159789423971876249376210418465005084962167362096100000562756870921473536902755883143961951665732630701252093626332547047170420815273463130790247802554687937658130710111
Private Key
Hello
Ciphertext
Creatinga RSA Public Key
Encryption
- e = mk mod(n)
Decryption
- m = edmod(n)
Where “n” is the product of two primes (pq) and “k” is greater than 1 and coprime to the totient of (pq)
Worked Example:
p = 61 and q = 53, hence “n” = 61 x 53 = 3233
The totient of the products of the primes is (61-1) x (53-1) = 3120
Now choose any number k > 1 which is coprime to 3120. We can choose 17. As 17 is not a factor of 3120
We have the public key components “n” and “k”, so we must now compute our private “d” component for decryption :
“d” is computed such that d multiplied by k mod (totient n) is congruent to 1, to do this we use our secret numbers “p”, “q” and the totient of “n”. Namely in this example 17 x d mod 3120 = 1.
Using back substitution in the Euclidean Theorem we find “d” to be 2753.
((15 x 3120) + 1 ) divided by 17 = 2753 or 17 x 2743 = 46801 and 46801 mod 3120 = 1

Cryptography

by gadgetic on Mar 19, 2010

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Cryptography — Presentation Transcript