Computer Science CPSC415 Biometrics and Cryptography1 Introduction to Public Key Cryptography Lecture 4.

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Introduction to

Public Key Cryptography

Lecture 4

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Outline

• Public Key Encryption

• Public Key Cryptographic System

• Public Key vs. Symmetric Key

• Digital Signatures

• Digital Envelope

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Insufficiencies with Symmetric Encryption

• Symmetric encryption is not enough to address two key issues– key distribution – how to have secure

communications in general without having to trust a KDC with your key?

– digital signatures – how to verify that a received message really comes from the claimed sender?

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Advent of Asymmetric Encryption

• Probably most significant advance in the 3000 year history of cryptography

• Use two keys: a public key and a private key

• Asymmetric since parties are not equal • Clever application of number theory

concepts instead of merely substitution and permutation

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How Asymmetric Encryption Works

• Asymmetric encryption uses two keys that are related to each other – a public key, which may be known to anybody, is

used to encrypt messages, and verify signatures – a private key, known only to the owner, is used to

decrypt messages encrypted by the matching public key, and create signatures

– the key used to encrypt messages or verify signatures cannot decrypt messages or create signatures

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Public key Encryption• Alice has a key pair: public and private

• publish the public key such that the key is publicly known• Alice keeps the private key secret

• Other people use Alice’s public key to encrypt messages for Alice

• Alice uses her private key to decrypt• Only Alice can decrypt since only Alice has the private key

Message Encrypt

Public key

Message Decrypt

Private key

• Trick: To compute the private key from the public key is a difficult problem.

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Asymmetric Encryptionfor Confidentiality

Bob Alice

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Asymmetric Encryptionfor Authentication

Bob Alice

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Applications for Asymmetric Encryption

• Three categories– Encryption/decryption: sender encrypts a

message with receiver’s public key– Digital signature: sender “signs” a message

with its private key– Key exchange: two sides exchange a

session key

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Security of Asymmetric Encryption

• Like symmetric schemes brute-force exhaustive search attack is always theoretically possible, but keys used are too large (>512bits)

• Not more secure than symmetric encryption, dependent on size of key

• Security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems

• Generally the hard problem is known, just made too hard to do in practice

• Require using very large numbers, so is slow compared to symmetric schemes

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Public key directory

encryption decryptionM

Alice’s public key

cryptanalysis Mprivate key

C MAlice

Eve

private key

Public key Cryptographic System

• C = EPK(M)

• M = DSK(C) = DSK(EPK(M))

• Public keys are published.• Each private key is known to the receiver only.• Difficult for Eve to find out SK from PK.

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Public key vs. Symmetric key

Symmetric key Public key

Both share same key(or one key is computable from the other)

Typically faster Typically slower

Two parties MUST trust each other

Two parties DO NOT need to trust each other

Two separate keys: a public and a private key

Examples:DES, IDEA, RC5, CAST, AES, …

Examples:RSA, ElGamal Encryption, ECC…

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Digital signatures

• Is there a functional equivalent to a handwritten signature?– Easy for legitimate user to sign– But hard for anyone else to forge– Easy for anyone to verify– Dependent on message & signer (key)

• Public key!– Sign: “invert” function using private key– Verify: compute function using public key

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Digital signatures

Message Sign

Private key

Message

Verify

Public key

Valid/Invalid

(fixed-length signature)

• Only the signer (who has a private key) can generate a valid signature

• Everyone (since the corresponding public key is published) can verify if a signature with respect to a message is valid

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Digital Envelopes-- Symmetric + Asymmetric

1. Generate a secret key (session key) at random.

2. Encrypt the message using the session key and symmetric algorithm.

3. Encrypt the session key with the recipient’s public key. This becomes the “digital envelope”.

4. Send the encrypted message and the digital envelope to the recipient.

• Figure …

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Plain Cipher Cipher Plain

Session Key Session Key

Session KeySession Key

Recipient’s

Public key

Digital Envelope

Digital Envelope

Digital Envelopes

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MotivationRevision• One problem with symmetric key algorithms is that

the sender needs a secure method of telling the receiver the key.

• Plus, you need a separate key for everyone you might communicate with.

• Public key algorithms use a public-key and private-key pair to tackle key management problem.– Each receiver has a public key pair.– The public key is publicly known (published).– A sender uses the receiver’s public key to encrypt a

message.– Only the receiver can decrypt it with the corresponding

private key.

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• Invented by Rivest, Shamir & Adleman of MIT in 1977 • Best known and widely used public-key scheme • Based on exponentiation in a finite (Galois) field over

integers modulo a prime – exponentiation takes O((log n)3) operations (easy)

• Use large integers (e.g. 1024 bits)• Security due to cost of factoring large numbers

– factorization takes O(e log n log log n) operations (hard)

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Computer Computer ScienceScienceRSA Key Setup

• Each user generates a public/private key pair by – select two large primes at random: p, q – compute their system modulus n=p·q

• note ø(n)=(p-1)(q-1) – select at random the encryption key e

• where 1<e<ø(n), gcd(e,ø(n))=1 – solve following equation to find decryption key d

• e·d=1 mod ø(n) and 0≤d≤n – publish their public encryption key: KU= {e,n} – keep secret private decryption key: KR= {d,n}

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Computer Computer ScienceScienceRSA Usage

• To encrypt a message M:– sender obtains public key of receiver KU={e,n} – computes: C=Me mod n, where 0≤M<n

• To decrypt the ciphertext C:– receiver uses its private key KR={d,n} – computes: M=Cd mod n

• Message M must be smaller than the modulus n (cut into blocks if needed)

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RSA Example: Computing Keys

1. Select primes: p=17, q=112. Compute n=pq=17×11=1873. Compute ø(n)=(p–1)(q-1)=16×10=1604. Select e: gcd(e,160)=1 and e<160

choose e=7

5. Determine d: de=1 mod 160 and d<160 d=23 since 23×7=161=10×160+1

6. Publish public key KU={7,187}7. Keep secret private key KR={23,187}

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RSA Example: Encryption and Decryption

• Given message M = 88 (88<187)

• Encryption KU={7,187} :C = 887 mod 187 = 11

• Decryption KR={23,187} :M = 1123 mod 187 = 88