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# Assignment: Public Key Cryptography Implementation

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CPE-321 Introduction of Computer Security
Assignment: Public Key Cryptography Implementation
Objectives
The objectives for this lab assignment are as follows:
• To explore public key cryptography
o Implementing the Diffie-Hellman Key Exchange protocol
o Implementing RSA encryption scheme
Background
In this lab, you will explore asymmetric key (public key) cryptography by
implementing the Diffie-Hellman Key Exchange protocol and RSA encryption
scheme. You will explore the properties of these schemes, and see how naive
implementations can lead to insecurity. Python and PyCrypto (or some other highlevel language with decent crypto support) are recommended to use for this lab.
accordingly:
Task 1: Implement Diffie-Hellman Key Exchange: The goal of this task is to get
your public key crypto juices flowing by implementing one of the single most
important discoveries in modern cryptography: Diffie Hellman Key Exchange. In a
single program, emulate the following protocol:
Alice Bob
Privately Computes Sends Privately Computes Sends
p,g
Pick random element a ∈ ℤp
A = ga mod p
Pick random element b ∈ ℤp
B = gb mod p
A B
s = Ba mod p s = Ab mod p
k = SHA256(s) k = SHA256(s)
m0 = “Hi Bob!”
c0 = AES-CBCk(m0)
c0 m1 = “Hi Alice!”
c1 = AES-CBCk(m1)
c1
Try it in a small group first, setting p=37 and g=5. Confirm Alice and Bob compute
the same symmetric key k. Truncate the output of SHA256 to 16 bytes, so that you
can use it to AES-CBC encrypt some messages between Alice and Bob.
Now, let’s use some “real life” numbers. IETF suggests the following 1024-bit
parameters:
p =
B10B8F96 A080E01D DE92DE5E AE5D54EC 52C99FBC FB06A3C6
9A6A9DCA 52D23B61 6073E286 75A23D18 9838EF1E 2EE652C0
13ECB4AE A9061123 24975C3C D49B83BF ACCBDD7D 90C4BD70
98488E9C 219A7372 4EFFD6FA E5644738 FAA31A4F F55BCCC0
A151AF5F 0DC8B4BD 45BF37DF 365C1A65 E68CFDA7 6D4DA708
DF1FB2BC 2E4A4371
g =
A4D1CBD5 C3FD3412 6765A442 EFB99905 F8104DD2 58AC507F
D6406CFF 14266D31 266FEA1E 5C41564B 777E690F 5504F213
160217B4 B01B886A 5E91547F 9E2749F4 D7FBD7D3 B9A92EE1
909D0D22 63F80A76 A6A24C08 7A091F53 1DBF0A01 69B6A28A
D662A4D1 8E73AFA3 2D779D59 18D08BC8 858F4DCE F97C2A24
855E6EEB 22B3B2E5
Modify your implementation to use these parameters, make sure Alice and Bob can
compute the same symmetric key k, and correctly exchange some encrypted
messages.
Task 2: Implement MITM key fixing & negotiated groups: Diffie-Hellman is
only secure against a passive (eavesdropping) adversary. Very bad things can happen
if the adversary is able to tamper with the numbers in the protocol.
Alice Mallory Bob
Computes Sends Modifies Computes Sends
p,g
a ∈ ℤp
A = ga mod p
b ∈ ℤp
B = gb mod p
A A→p
B→p B
s = Ba mod p s = Ab mod p
k = SHA256(s) k = SHA256(s)
m0 = “Hi Bob!”
c0 = AESCBCk(m0)
c0 m1 = “Hi Alice!”
c1 = AESCBCk(m1)
c1
2). Repeat this attack, but instead of tampering with A and B, tamper with the
generator g. Show that Mallory can recover the message 𝑐0 𝑎𝑛𝑑 𝑐1 by setting g to 1,
p, or p-1.
Task 3: Implement “textbook” RSA & MITM Key Fixing via Malleability:
1). RSA has two core components: key generation and encryption/decryption. (90%
of the work is in implementing key generation.) Your implementation should support
variable length primes (up to 2048 bits), and use the value e=65537. Feel free to use
your cryptographic library’s interface for generating large primes, but implement the
rest-including computing the multiplicative inverse – yourself.
Encrypt and decrypt a few messages to yourself to make sure it works. Remember
messages must be integers in 𝑍𝑛

. You can convert an ASCII string to hex, and then
turn that hex value into an integer.
2). From 1). you just implemented “textbook” RSA, and it is widely insecure.
Because it is too slow and inconvenient to operate on a large amount data directly,
RSA is often used to exchange a symmetric key that will be used to encrypt future
messages. It would be terrible, of course, if an adversary were able to learn that key.
And, that’s what we’re about to do. One of textbook RSA’s great weaknesses is its
malleability, i.e. an active attacker can change the meaning of the plaintext message
by performing an operation on the respective ciphertext. To demonstrate the dangers
of malleability, implement the following protocol:
Alice Mallory Bob
Computes Sends Modifies Computes Sends
n,e
s ∈ ℤ
*
n
c = se mod n
c
c’ = F(c)
s = c’d mod n
k = SHA256(s)
m = “Hi Bob!”
c0 = AES-CBCk(m)
c0
Find the operation F() that Mallory needs to apply to the ciphertext c that will allow
her to decrypt the ciphertext c0. Hint: Mallory knows Alice’s public key (n,e) and can
encrypt her own messages to Alice to allow Mallory to know the value of s.
Give another example of how RSA’s malleability could be used to exploit a system
(e.g. to cause confusion, disruption, or violate integrity).
Malleability can also affect signature schemes based on RSA. Consider the scheme:
Sign(m,d) = 𝑚𝑑mod n
Suppose Mallory sees the signatures for two messages 𝑚1 𝑎𝑛𝑑 𝑚2. Show how
Mallory can create a valid signature for a third message, 𝑚3 = 𝑚1 ∙ 𝑚2.
Alice Bob
Privately Computes Sends Privately Computes Sends
p,g
a ∈ ℤp
A = ga mod p
b ∈ ℤp
B= gb mod p
A B
s = Ba mod p
Forget a
s = Ab mod p
Forget b
k = SHA256(s) k = SHA256(s)
m0 = “Hi Bob!”
c0 = AES-CBCk(m0)
c0 m1 = “Hi Alice!”
c1 = AES-CBCk(m1)
c1
Protocol 1: A Diffie-Hellman Key-agreement
Alice Bob
Privately Computes Sends Privately Computes Sends
RSA key pair <Apub,Aprii>
Apub
Pick random element x ∈ ℤ
*
n
y = RSA(Apub,x)
y
x = RSA-1
(Apri, y)
k = SHA256(x) k = SHA256(x)
m0 = “Hi Bob!”
c0 = AES-CBCk(m0)
c0 m1 = “Hi Alice!”
c1 = AES-CBCk(m1)
c1
Protocol 2: An RSA Key Exchange
Questions:
1. For task 1, how hard would it be for an adversary to solve the Diffie Hellman
Problem (DHP) given these parameters? What strategy might the adversary
take?
2. For task 1, would the same strategy used for the tiny parameters work for the
p and g? Why or why not?
3. For task 2, why were these attacks possible? What is necessary to prevent it?
4. For task 3 part 1, while it’s very common for many people to use the same
value for e in their key (common values are 3, 7, 216+1), it is very bad if two
people use the same RSA modulus n. Briefly describe why this is, and what
the ramifications are.
Submission: Submit a report describing what you did and what you observed.
Include any code that you wrote, as well as answers to any questions (there are in
italics). Please include any explanations of the surprising or interesting observations