**Blum–Goldwasser Cryptosystem**

The **Blum-Goldwasser** (BG) cryptosystem is an asymmetric key encryption algorithm proposed by Manuel Blum and Shafi Goldwasser in 1984. Blum-Goldwasser is a probabilistic, semantically secure cryptosystem with a constant-size ciphertext expansion. The encryption algorithm implements an XOR-based stream cipher using the Blum Blum Shub (BBS) pseudo-random number generator to generate the keystream. Decryption is accomplished by manipulating the final state of the BBS generator using the secret key, in order to find the initial seed and reconstruct the keystream.

The BG cryptosystem is semantically secure based on the assumed intractability of integer factorization; specifically, factoring a composite value where are large primes. BG has multiple advantages over earlier probabilistic encryption schemes such as the Goldwasser-Micali cryptosystem. First, its semantic security reduces solely to integer factorization, without requiring any additional assumptions (e.g., hardness of the quadratic residuosity problem or the RSA problem). Secondly, BG is efficient in terms of storage, inducing a constant-size ciphertext expansion regardless of message length. BG is also relatively efficient in terms of computation, and fares well even in comparison with cryptosystems such as RSA (depending on message length and exponent choices). However, BG is highly vulnerable to adaptive chosen ciphertext attacks (see below).

Because encryption is performed using a probabilistic algorithm, a given plaintext may produce very different ciphertexts each time it is encrypted. This has significant advantages, as it prevents an adversary from recognizing intercepted messages by comparing them to a dictionary of known ciphertexts.

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**Blum–Goldwasser Cryptosystem**- Security and Efficiency

... Depending on plaintext size, BG may be more or less computationally expensive than RSA ... Because most RSA deployments use a fixed encryption exponent optimized to minimize encryption time, RSA encryption will typically outperform BG for all but the shortest messages ...