# CS代考程序代写 scheme algorithm Cryptography Basics – Public Key Cryptography

Cryptography Basics – Public Key Cryptography

ECEN 4133 FEB 4, 2021

Shared key limitations

Suppose Alice publishes data to lots of people, and they all want to verify integrity… Can’t share an integrity key with everybody, or else anybody could forge messages

Suppose Bob wants to receive data from lots of people, confidentially…

Schemes we’ve discussed would require a separate key shared with each person

[What to do?]

Public-key crypto

So far, encryption key == decryption key “symmetric key crypto” New idea: Keys are distinct, and you can’t find one from the other

Almost always used by splitting key-pair Alice keeps one key private (“private key”)

Publishes the other key (“public key”)

Invented in 1976 by Diffie and Hellman (earlier by Clifford Cocks of British intelligence, in secret)

First popular public key algorithm: RSA Rivest, Shamir, and Adleman 1978

Requirements for a public key crypto system to be secure

1. Computationally easy for B to generate a key pair: PUb, PRb

2. Computationally easy for sender A to generate the ciphertext for message M: C=E(PUb, M)

3. Computationally easy for receiver B to decrypt the ciphertext: M=D(PRb, C)

4. Computational infeasible to guess PRb knowing PUb.

5. Computational infeasible to recover M from PUb and C.

RSA

How RSA works

Key generation:

1. Pick large (say, 1024 bits) random primes p and q

2. Compute N := pq

(RSA uses multiplication mod N)

3. Pick e to be relatively prime to (p-1)(q-1)

4. Find d so that ed mod (p-1)(q-1) = 1

5. Finally: Public key is (e,N) Privatekeyis (d,N)

e

To encrypt: E(x) = x mod N

Todecrypt: D(x)=xdmodN

Why RSA works

“It works” theorem: For all 0 < x < N,
can show that D(E(x)) = x Proof:
D(E(x)) = (xe mod pq)d mod pq = xed mod pq
= xa(p-1)(q-1)+1 mod pq for some a = (x(p-1)(q-1))ax mod pq
= (x(p-1)(q-1) mod pq)ax mod pq = 1ax mod pq
(because of the fact that if p,q x(p-1)(q-1) mod pq = 1)
=x
(because ed mod (p-1)(q-1) = 1)
are prime, then for all 0 < x < N,
Is RSA secure?
Best known way to compute d from e is factoring N into p and q.
Best known factoring algorithm:
General number field sieve
Takes more than polynomial time, but less than exponential time, to factor n-bit number. (Still takes way too long if p,q are large enough and random.)
Fingers crossed...
but can’t rule out a breakthrough!
Signing with the public key for confidentiality or secrecy:
Does this provide integrity?
Signing with private key for integrity/authentication.
Does this provide confidentiality?
RSA can be used for either confidentiality or integrity
RSA for confidentiality:
Encrypt with public key Decrypt with private key
“your eyes only” message
RSA for integrity:
Encrypt (“sign”) with private key Decrypt (“verify”) with public key
called a digital signature
[What if we want both confidentiality and integrity on the same message?]
Which of these provides both confidentiality and integrity?
Alice (A) wants to send a secret message M to Bob (B) so that Bob can verify that it comes from Alice.
Which one(s) is/are secure?
1. E(E(M, PRA), PUB)
2. E(E(M, PUB), PRA)
3. C=E(M, PRA) t=E(H(C), PUB)
◦ Send C||t
4. C=E(M, PUB) t=E(H(C), PRA) ◦ Send C||t
Review: Public-key Crypto So far, encryption key == decryption key
“symmetric key crypto” New idea: Keys are distinct.
RSA: N := pq
Public key is (e,N)
Private key is (d,N) To encrypt: E(x)
To decrypt: D(x) RSA for confidentiality:
Encrypt with public key Decrypt with private key
= xe mod N = xd mod N
RSA for integrity (digital signatures): Encrypt (“sign”) with private key
Decrypt (“verify”) with public key
[Cautions?!]
RSA drawback: Performance
Factor of 1000 or more slower than AES.
Dominated by exponentiation – cost goes up (roughly) as cube of key size.
Message must be shorter than N. [How big should the RSA keys be?]
Use in practice:
Encryption:
Use RSA to encrypt a random x < N, compute k := PRF(x), encrypt message using a symmetric cipher and key k
Signing:
Compute v := PRF(m), use RSA to sign a carefully padded version of v (many gotchas!)
Almost always should use crypto libraries to get the details right
True or False?
Public-key encryption is a general-purpose technique that has made symmetric encryption obsolete
True or False?
Key distribution is trivial when using public-key encryption, compared to the cumbersome handshaking involved with key distribution centers for symmetric encryption.
Attacks against RSA
1. Brute force: trying all possible private keys
2. Mathematical attacks: factoring
3. Timing attacks: using the running time of decryption
4. Hardware-based fault attack: induce faults in hardware to generate digital signatures
5. Chosen plaintext attack on unpadded RSA
Exercise
Suppose Bob uses RSA crypto with a very large modulus n for which the factorization cannot be found in a reasonable amount of time.
Suppose Alice sends a message to Bob by representing each alphabet letter as an integer between 0 and 25 (A->0, …, Z->25) and then encrypting each number separately using RSA with large e and large n.

Is this method secure?

If yes, why?

If not, how to efficiently attack this encryption method?

Solution

For a set of message block values SM = {0, 1, 2, …, 25}. The set of corresponding ciphertext block values SC = {0e mod N, 1e mod N, …, 25e mod N}, and can be computed by everybody with the knowledge of the public key of Bob.

The most efficient attack is to compute Me mod N for all possible values of M, then create a look-up table with a ciphertext as an index, and the corresponding plaintext as a value of the appropriate location in the table.

Two subtle “textbook” RSA problems:

1. For small e and m: m^e mod N == m^e Trivial to decrypt!

2. If m is chosen from a small set, easy to confirm a ciphertext is a given message (anyone can encrypt!)

Chosen plaintext attack

Solution: RSA Padding

Need to make sure m is as large enough to wrap around N (so can’t take e-th root of ciphertext) Need to randomize before encryption (so low-entropy plaintext can’t be decrypted)

Other public key cryptography systems

RSA is popular, but not the only one:

◦ DSA – Digital Signature Algorithm

◦ ECDSA – Elliptic Curve Digital Signature Algorithm ◦ Very small public keys: e.g. curve25519: 256-bits (32 bytes)

◦ Post-Quantum Cryptography: ◦ Ring-LWE, NTRU, hash-based

So Far:

The Security Mindset Message Integrity Confidentiality

Key Exchange

Building a Secure Channel Public Key Crypto

Next Week:

Begin Web Security Unit

HTTPS: Secure channels for the web