Data assimilation

The Goldwasser–Micali (GM) cryptosystem is an asymmetric key encryption algorithm developed by Shafi Goldwasser and Silvio Micali in 1982. GM has the distinction of being the first probabilistic public-key encryption scheme which is provably secure under standard cryptographic assumptions. However, it is not an efficient cryptosystem, as ciphertexts may be several hundred times larger than the initial plaintext. To prove the security properties of the cryptosystem, Goldwasser and Micali proposed the widely used definition of semantic security.

Basis

The GM cryptosystem is semantically secure based on the assumed intractability of the quadratic residuosity problem modulo a composite N = pq where p, q are large primes. This assumption states that given (x, N) it is difficult to determine whether x is a quadratic residue modulo N (i.e., x = y² mod N for some y), when the Jacobi symbol for x is +1. The quadratic residue problem is easily solved given the factorization of N, while new quadratic residues may be generated by any party, even without knowledge of this factorization. The GM cryptosystem leverages this asymmetry by encrypting individual plaintext bits as either random quadratic residues or non-residues modulo N, all with quadratic residue symbol +1. Recipients use the factorization of N as a secret key, and decrypt the message by testing the quadratic residuosity of the received ciphertext values.

Because Goldwasser–Micali produces a value of size approximately |N| to encrypt every single bit of a plaintext, GM encryption results in substantial ciphertext expansion. To prevent factorization attacks, it is recommended that |N| be several hundred bits or more. Thus, the scheme serves mainly as a proof of concept, and more efficient provably-secure schemes such as Elgamal have been developed since.

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.

Scheme definition

Goldwasser–Micali consists of three algorithms: a probabilistic key generation algorithm which produces a public and a private key, a probabilistic encryption algorithm, and a deterministic decryption algorithm.

The scheme relies on deciding whether a given value x is a square mod N, given the factorization (p, q) of N. This can be accomplished using the following procedure:

1. Compute x_p = x mod p, x_q = x mod q.
2. If $x_{p}^{(p-1)/2}\equiv 1{\pmod {p}}$ and $x_{q}^{(q-1)/2}\equiv 1{\pmod {q}}$, then x is a quadratic residue mod N.

Key generation

The modulus used in GM encryption is generated in the same manner as in the RSA cryptosystem. (See RSA, key generation for details.)

1. Alice generates two distinct large prime numbers p and q, randomly and independently of each other.
2. Alice computes N = p q.
3. She then finds some non-residue x such that the Legendre symbols satisfy $\left({\frac {x}{p}}\right)=\left({\frac {x}{q}}\right)=-1$ and hence the Jacobi symbol $\left({\frac {x}{N}}\right)$ is +1. The value x can for example be found by selecting random values and testing the two Legendre symbols. If p, q = 3 mod 4 (i.e., N is a Blum integer), then the value N − 1 is guaranteed to have the required property.

The public key consists of (x, N). The secret key is the factorization (p, q).

Message encryption

Suppose Bob wishes to send a message m to Alice:

1. Bob first encodes m as a string of bits (m₁, ..., m_n).
2. For every bit $m_{i}$, Bob generates a random value $y_{i}$ from the group of units modulo N, or $\gcd(y_{i},N)=1$. He outputs the value $c_{i}=y_{i}^{2}x^{m_{i}}{\pmod {N}}$.

Bob sends the ciphertext (c₁, ..., c_n).

Message decryption

Alice receives (c₁, ..., c_n). She can recover m using the following procedure:

1. For each i, using the prime factorization (p, q), Alice determines whether the value c_i is a quadratic residue; if so, m_i = 0, otherwise m_i = 1.

Alice outputs the message m = (m₁, ..., m_n).

Security properties

There is a simple reduction from breaking this cryptosystem to the problem of determining whether a random value modulo N with Jacobi symbol +1 is a quadratic residue. If an algorithm A breaks the cryptosystem, then to determine if a given value x is a quadratic residue modulo N, we test A to see if it can break the cryptosystem using (x,N) as a public key. If x is a non-residue, then A should work properly. However, if x is a residue, then every "ciphertext" will simply be a random quadratic residue, so A cannot be correct more than half of the time. Furthermore, this problem is random self-reducible, which ensures that for a given N, every public key is just as secure as every other public key.

The GM cryptosystem has homomorphic properties, in the sense that if c₀, c₁ are the encryptions of bits m₀, m₁, then c₀c₁ mod N will be an encryption of $m_{0}\oplus m_{1}$. For this reason, the GM cryptosystem is sometimes used in more complex cryptographic primitives.

References

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

See also

• Blum–Goldwasser cryptosystem