Benaloh cryptosystem


The Benaloh Cryptosystem is an extension of the Goldwasser-Micali cryptosystem created in 1985 by Josh Benaloh. The main improvement of the Benaloh Cryptosystem over GM is that longer blocks of data can be encrypted at once, whereas in GM each bit is encrypted individually.

Scheme Definition

Like many Public [key cryptography|public key cryptosystems], this scheme works in the group where n is a product of two large primes. This scheme is homomorphic and hence malleable.

Key Generation

Given block size r, a public/private key pair is generated as follows:
  1. Choose large primes p and q such that and
  2. Set
  3. Choose such that.
  4. Set
The public key is then, and the private key is.

Message Encryption

To encrypt message :
  1. Choose a random
  2. Set

Message Decryption

To decrypt a ciphertext :
  1. Compute
  2. Output, i.e., find m such that
To understand decryption, first notice that for any and we have:
To recover m from a, we take the discrete log of a base x. If r is small, we can recover m by an exhaustive search, i.e. checking if for all. For larger values of r, the Baby-step giant-step algorithm can be used to recover m in time and space.

Security

The security of this scheme rests on the Higher residuosity problem, specifically, given z,''r and n'' where the factorization of n is unknown, it is computationally infeasible to determine whether z is an rth residue mod n, i.e. if there exists an x such that.