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The Benaloh Cryptosystem is an extension of the Goldwasser-Micali cryptosystem (GM) created in 1994 by Josh (Cohen) 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. Contents 1 Scheme Definition 1.1 Key Generation 1.2 Message Encryption 1.3 Message Decryption 1.4 Security 2 References Scheme Definition Like many 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 A public/private key pair is generated as follows: Choose a blocksize r. Choose large primes p and q such that r divides (p-1) and gcd(q-1,r) = 1. Set n = pq Choose such that . The public key is then y,n, and the private key is the two primes p,q. Message Encryption To encrypt a message m, where m is taken to be an element in Choose a random Set Message Decryption To understand decryption, we first notice that for any m,u we have Since m < r and , we can conclude that if and only if m = 0. So if is an encryption of m, given the secret key p,q we can determine whether m=0. If r is small, we can decrypt z by doing an exhaustive search, i.e. decrypting the messages y-iz for i from 1 to r. By precomputing values, using the Baby-step giant-step algorithm, decryption can be done in time . Security The security of this scheme rests on an extension of the Quadratic 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 . References Original Paper (ps) v · d · e Public-key cryptography Algorithms Benaloh · Blum–Goldwasser · Cayley–Purser · CEILIDH · Cramer–Shoup · Damgård–Jurik · DH · DSA · EPOC · ECDH · ECDSA · EKE · ElGamal (encryption · signature scheme) · GMR · Goldwasser–Micali · HFE · IES · Lamport · McEliece · Merkle–Hellman · MQV · Naccache–Stern · NTRUEncrypt · NTRUSign · Paillier · Rabin · RSA · Okamoto–Uchiyama · Schnorr · Schmidt–Samoa · SPEKE · SRP · STS · Three-pass protocol · XTR Theory Discrete logarithm · Elliptic curve cryptography · RSA problem Standardization ANS X9F1 · CRYPTREC · IEEE P1363 · NESSIE · NSA Suite B Topics Digital signature · OAEP · Fingerprint · PKI · Web of trust · Key size v · d · e Cryptography History of cryptography · Cryptanalysis · Cryptography portal · Outline of cryptography Symmetric-key algorithm · Block cipher · Stream cipher · Public-key cryptography · Cryptographic hash function · Message authentication code · Random numbers · Steganography