Question

Cyber Security class

1. Your task is to compute a session key KAB in Diffie-Hellman Key Exchange (DHKC) with Elliptic Curves. Your private key is a = 6. You receive Bob’s public key B = (5,9). The elliptic curve being used is defined by y2 ≡ x3+x+6 mod 11.

2. In RSA Digital Signature, Suppose Bob wants to send a signed message (x = 4) to Alice. The first steps are exactly the same as it is done for an RSA encryption: Bob computes his RSA parameters and sends the public key to Alice. We know p = 3, q =11, and bob choose e=3. (Hint: We learn RSA algorithm and key generation in Week 7)

(a) What is the public key pair Bob sends to Alice?

(b) What is the value of signature s?

(c) What is the value of verkpubA(x,s)? Show all intermediate steps clearly.

Prove in RSA digital signature, verkpubA(x,s) = x

Answer 1

1 Ans)
Given Curve y² = x³ + x + 6

bob's Public Key : (5, 9)

Alice Private Key : 6

Shared Key : bob_pub_key * Alice_priv_key on given Eliptical curve field

I have used sagemath (opensource) to do this

Shared Key : (2, 7)  

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2 Ans)

Given p = 11 , q = 3 # Primes used for modulus

n = p*q = 33 # modulus

e = 3 # Public Key

totient = (p-1) * (q - 1) = 10*2 = 20

d = 3^-1 % 20 = 7 # 3*7 %20 = 21 % 20 = 1 # Private Key

x = 4 # message be 4

a)  public key pair Bob sends to Alice is (n, e) = (33, 3)  

b) Siganture (m^d %n) = (4⁷  % 33) = 16  

c) Verify the Signature

(sig^e % n) = msg # To verify a siganture of msg as (n, e) is public key and it is shared

(16³ % 33) = (4096 % 33) = 4 # Message