Cryptographic Protocols - Science and Maths Assignment Help

Assignment Task

1.1 A flawed attempt of semantically secure RSA scheme 

Consider the following public key encryption scheme £ = (G, E, D):

G(-): Same as RSA, namely given a (public) odd integer e, and a parameter size £, generate
p,q prime numbers of £ bits such that ged(e,p — 1) = 1, ged(e,qg—1) = 1.

Compute N =p-q,o(N)=(p—1)-(¢g—1) and d =e! mod p(N).

Output pk = (N, e) as public key and sk = (N, d) as private key.

The message space is M = Z%, and the ciphertext space is C = (Z%)?

e E(pk,m): Choose a random z € Z};, define

ci =z mod N and coc=xz-m mod N and output the ciphertext ¢ = (e1, 2)

D(sk,c): Compute Mm = cp/cf mod N and output 7.

(a) Show it is indeed a public key encryption scheme, namely it satisfies that for all m € M, D(sk, E(pk,m)) =m

(b) Show £ is NOT semantically secure.

1.2 A modification of multiplicative El Gamal public key encryption scheme

Let G = {g) be a group of prime order ¢ generated by g. Consider the following public key encryp-
tion scheme over message and ciphertext spaces M = G and C = G3.

• G(-): Chooses a1, at random in Zj = {1,...,q — 1} and defines u; = g™*,up = g**. The public key is pk = (u1, ug) and the secret key is sk = (a1, a2).

• E(pk,m): It chooses random $; and B2 in Z; = {0,...,q — 1} and outputs the following ciphertext (co,c1, 2) = (gm, uf, uf?)

•  D(sk,(co,c1,¢2)): ?

(a)  Find an efficient deterministic decryption algorithm D.

(b) Prove the scheme is CPA-secure if the DDH (the decisional Diffie-Hellman) assumption holds for G. (4 points)

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