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OUTLINEIntroductionGoals In CryptographySecrete Key CryptographyPublic Key CryptograpgyDigital Signatures

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AN OVERVIEW OF CRYPTOGRAPHY:

Cryptography is derived from the Greek words: Kryptos, “hidden”, and Graphein, ”to write ” or “hidden writing”.

The word cryptography means “secret writing”. However, the term today refers to the science and of transforming messages to make them secure and immune to attacks.

The original message before being transformed is called plaintext. After the message is transformed, it is called cipher text. An encryption algorithm transforms the plaintext to cipher; a decryption algorithm transforms the cipher text back to plaintext. The sender uses an encryption algorithm, and the receiver uses a decryption algorithm.

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Goals in Cryptography Message Confidentiality Message integrity Sender authentication

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5

Encryption Decryption

SENDER RECEIVER

NetworkCiphertext Ciphertext

Plaintext

Cryptography Components

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  These encryption and decryption algorithms are called as ciphers (categories of algorithm). One cipher can serve millions of communicating pairs. A Key is value that the cipher, as an algorithm, operates on. To encrypt a message we need an encryption algorithm, an encryption key, and the plain text. These create the cipher text. To decrypt a message, we need a decryption algorithm, and the cipher text. So these reveal the original plaintext.

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ENCRYPTION KEY

ENCRYPTION ALGORITHM

DECRYPTION ALGORITHM

DECRYPTION KEY

Plain text

Ciphertext

Plain text

a. Encryption b. Decryption

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Secret Key Cryptography•Single key used to encrypt and decrypt.

•Key must be known by both parties.

•Assuming we live in a hostile environment (otherwise - why the need for cryptography?), it may be hard to share a secret key.

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In Cryptography, the encryption/decryption algorithms are public; anyone can access them. The keys are secret. So they need to be protected.

Cryptography algorithms can be divided into two groups.

Symmetric-key cryptography (or secret key) algorithmPublic-key cryptography (or asymmetric key) algorithm

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Symmetric-key cryptographyIf the sender and recipient must have the

same key in order to encode or decode the protected information , then the cipher is a symmetric key cipher since everyone uses the same key for the same message . The main problem is that the secret key must somehow be given to both the sender and recipient privately. For this reason, symmetric key (or secret key ) ciphers.

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Public Key Cryptography(a.k.a. asymmetric cryptography)• If the sender and recipient have different

keys respective to the communication roles they play, then the cipher is an asymmetric key cipher as different keys exist for encoding and decoding the same message.

• Each entity has 2 keys:› private key (a secret)› public key (well known).

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•Private keys are used for decrypting.

•Public keys are used for encrypting.

encryptionplaintext ciphertext

public key

decryptionciphertext plaintext

private key

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Using Keys

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Digital Signature•Public key cryptography is also used to

provide digital signatures.

signingplaintext signed message

private key

verificationsigned message plaintext

public key

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AliceAlice BobBob

Sign with Aprivate check signature using Apublic

encrypt using Bpublic decrypt using Bprivate

Revised Scheme

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RSA AlgorithmIn cryptography, RSA (which stands for Rivest, Shamir and Adleman who first publicly described it) is an algorithm for public-key cryptography.[1] It is the first algorithm known to be suitable for signing as well as encryption, and was one of the first great advances in public key cryptography. RSA is widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys and the use of up-to-date implementations.

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Operation The RSA algorithm involves three steps: key generation,

encryption and decryption.

Key generationRSA involves a public key and a private key. The public key can be known to everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted using the private key. The keys for the RSA algorithm are generated the following way:

1. Choose two distinct prime numbers p and q. For security purposes, the integers p and q should be chosen

uniformly at random and should be of similar bit-length. Prime integers can be efficiently found using a primality test.

2. Compute n = pq. n is used as the modulus for both the public and private key3. Compute φ(pq) = (p − 1)(q − 1). (φ is Euler's totient function).4. Choose an integer e such that 1 < e < φ(pq), and e and φ(pq)

share no divisors other than 1 (i.e., e and φ(pq) are coprime).

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e is released as the public key exponent. e having a short bit-length and small Hamming weight results in

more efficient encryption. However, small values of e (such as e = 3) have been shown to be less secure in some settings.

5. Determine d (using modular arithmetic) which satisfies the congruence relation .

Stated differently, ed − 1 can be evenly divided by the totient (p − 1)(q − 1).

This is often computed using the extended Euclidean algorithm. d is kept as the private key exponent.

The public key consists of the modulus n and the public (or encryption) exponent e. The private key consists of the private (or decryption) exponent d which must be kept secret.

Note:An alternative, used by PKCS#1, is to choose d matching e d ≡ 1 (mod λ) with λ = lcm(p-1,q-1), where lcm is the least common multiple. Using λ instead of φ(n) allows more choices for d. λ can also be defined using the Carmichael function λ(n).

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An alternative, used by PKCS#1, is to choose d matching e d ≡ 1 (mod λ) with λ = lcm(p-1,q-1), where lcm is the least common multiple. Using λ instead of φ(n) allows more choices for d. λ can also be defined using the Carmichael function λ(n).

For efficiency the following values may be precomputed and stored as part of the private key: p and q: the primes from the key generation, and

EncryptionAlice transmits her public key (n,e) to Bob and keeps the private key secret. Bob then wishes to send message M to Alice.He first turns M into an integer 0 < m < n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext c corresponding to:

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This can be done quickly using the method of exponentiation by squaring. Bob then transmits c to Alice.

Decryption

Alice can recover m from c by using her private key exponent d by the following computation:

Given m, she can recover the original message M by reversing the padding scheme.

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