Rsa

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In cryptography, RSA is an algorithm for public-key cryptography. It was the first algorithm known to be suitable for digital signature as well as encryption, and one of the first great advances in public key cryptography. RSA is widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys and the use of up-to-date implementations.

History The algorithm was publicly described in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman at Massachusetts Institute of Technology; the letters RSA are the initials of their surnames. Apocryphally , it was invented at a Passover Seder in Schenectady, New York SIAM News, Volume 36, Number 5, June 2003, "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders", by Sara Robinson

Clifford Cocks, a British mathematician working for the United Kingdom intelligence agency Government Communications Headquarters, described an equivalent system in an internal document in 1973, but given the relatively expensive computers needed to implement it at the time, it was mostly considered a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification, and Rivest, Shamir, and Adleman devised RSA independently of Cocks' work.

Massachusetts Institute of Technology was granted for a "Cryptographic communications system and method" that used the algorithm in 1983. The patent expired on 21 September 2000. Since a paper describing the algorithm had been published in August 1977, prior to the December 1977 filing date of the patent application, regulations in much of the rest of the world precluded patents elsewhere and only the United States patent was granted. Had Cocks' work been publicly known, a patent in the US would not have been possible either.

Operation RSA involves a public Key (cryptography) and a private key. The public key can be known to everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted using the private key. The keys for the RSA algorithm are generated the following way:

Choose two distinct large random prime numbers p \, and q \,

Compute n = p q \,

n\, is used as the Modular arithmetic for both the public and private keys

Compute the totient: \phi(n) = (p-1)(q-1) \,.

Choose an integer e\, such that 1 < e\, < \phi(n)\,, and e\, and \phi (n)\, share no factors other than 1 (coprime)

e\, is released as the public key exponent

Compute d\, to satisfy the Modular arithmetic#The congruence relation d e \equiv 1\pmod{\phi(n)}\,; i.e. de = 1 + k\phi(n)\, for some integer k\,.

d\, is kept as the private key exponent

Notes on the above steps:

Step 1: Numbers can be Primality test#Probabilistic_tests for primality.
Step 3: changed in PKCS#1 v2.0 to \lambda(n) = {\rm lcm}(p-1, q-1) \, instead of \phi(n) = (p-1)(q-1) \,.
Step 4: A popular choice for the public exponents is e\, = 216 + 1 = 65537. Some applications choose smaller values such as e\, = 3, 5, or 35 instead. This is done to make encryption and signature verification faster on small devices like smart cards but small public exponents may lead to greater security risks.
Steps 4 and 5 can be performed with the extended Euclidean algorithm; see modular arithmetic.

The public key consists of the modulus n\, and the public (or encryption) exponent e\,.
The private key consists of the modulus n\, and the private (or decryption) exponent d\, which must be kept secret.

For efficiency a different form of the private key can be stored:

p\, and q\,: the primes from the key generation,
d\mod (p - 1)\, and d\mod(q - 1)\,: often called dmp1 and dmq1.
q^{-1} \mod(p)\,: often called iqmp

All parts of the private key must be kept secret in this form. p\, and q\, are sensitive since they are the factors of n\,, and allow computation of d\, given e\,. If p\, and q\, are not stored in this form of the private key then they are securely deleted along with other intermediate values from key generation.
Although this form allows faster decryption and signing by using the Chinese Remainder Theorem, it is considerably less secure since it enables side channel attacks. This is a particular problem if implemented on smart cards, which benefit most from the improved efficiency. (Start with y = x^e mod n and let the card decrypt that. So it computes y^d \pmod{p} or y^d \pmod{q} whose results give some value z. Now, induce an error in one of the computations. Then \gcd(z-x,n) will reveal p or q.)

Encrypting messages Alice transmits her public key (n\, & e\,) to Bob and keeps the private key secret. Bob then wishes to send message M to Alice.

He first turns M into a number m\, < n\, by using an agreed-upon reversible protocol known as a #Padding schemes. He then computes the ciphertext c\, corresponding to:

c = m^e \mod{n}

This can be done quickly using the method of exponentiation by squaring. Bob then transmits c\, to Alice.

Decrypting messages Alice can recover m\, from c\, by using her private key exponent d\, by the following computation:

m = c^d \mod{n}.

Given m\,, she can recover the original message M.

The above decryption procedure works because first

c^d \equiv (m^e)^d \equiv m^{ed}\pmod{n}.

Now, e d \equiv 1\pmod{(p - 1)(q - 1)}, and hence

e d \equiv 1\pmod{p - 1}\, and e d \equiv 1\pmod{q - 1}\,

which can also be written as

e d = k (p - 1) + 1\, and e d = h (q - 1) + 1\,

for proper values of k\, and h\,. If m\, is not a multiple of p\, then m\, and p\, are coprime because p\, is prime; so by Fermat's little theorem

m^{(p-1)} \equiv 1 \pmod{p}

and therefore, using the first expression for e d\,,

m^{ed} = m^{k (p-1) + 1} = (m^{p-1})^k m \equiv {1}^k m = m \pmod{p}\,.

If instead m\, is a multiple of p\,, then

m^{ed} \equiv 0^{ed} = 0 \equiv m \pmod{p}.

Using the second expression for e d\,, we similarly conclude that

m^{ed} \equiv m \pmod{q}\,.

Since p\, and q\, are distinct prime numbers, applying the Chinese remainder theorem to these two congruences yields

m^{ed} \equiv m \pmod{pq}.

Thus,

c^d \equiv m \pmod{n}.

A working example Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but you can also Wikibooks:Transwiki:Generate a keypair using OpenSSL.

Choose two prime numbers

:p = 61 and q=53

Compute n = p q \,

:n=61*53=3233

Compute the totient \phi(n) = (p-1)(q-1) \,

:\phi(n) = (61 - 1)(53 - 1) = 3120\,

Choose e>1 coprime to 3120

:e=17

Compute d\, such that d e \equiv 1\pmod{\phi(n)}\, (d is uniquely determined by e and \phi(n)\,)

:d=2753

:17 * 2753 = 46801 = 1 + 15 * 3120.

The public key is (n=3233, e=17). For a padded message m\, the encryption function is:

c = m^e\mod {n} = m^{17} \mod {3233}.

The private key is (n=3233, d=2753). The decryption function is:

m = c^d\mod {n} = c^{2753} \mod {3233}.
For example, to encrypt m=123, we calculate

c = 123^{17}\mod {3233} = 855.

To decrypt c = 855, we calculate

m = 855^{2753}\mod {3233} = 123.

Both of these calculations can be computed efficiently using the square-and-multiply algorithm for modular exponentiation.

Padding schemes When used in practice, RSA is generally combined with some padding (cryptography). The goal of the padding scheme is to prevent a number of attacks that potentially work against RSA without padding:

When encrypting with low encryption exponents (e.g., e = 3) and small values of the m, (i.e. m

References
- Menezes, Alfred; van Oorschot, Paul C.; and Vanstone, Scott A. Handbook of Applied Cryptography. CRC Press, October 1996. ISBN 0-8493-8523-7
- R. Rivest, A. Shamir, L. Adleman. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM, Vol. 21 (2), pp.120–126. 1978. Previously released as an MIT "Technical Memo" in April 1977. Initial publication of the RSA scheme.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.7: The RSA public-key cryptosystem, pp.881–887.
External links
- PKCS #1: RSA Cryptography Standard (RSA Laboratories website)
In cryptography, RSA is an algorithm for public-key cryptography. It was the first algorithm known to be suitable for digital signature as well as encryption, and one of the first great advances in public key cryptography. RSA is widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys and the use of up-to-date implementations.

History The algorithm was publicly described in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman at Massachusetts Institute of Technology; the letters RSA are the initials of their surnames. Apocryphally , it was invented at a Passover Seder in Schenectady, New York SIAM News, Volume 36, Number 5, June 2003, "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders", by Sara Robinson

Clifford Cocks, a British mathematician working for the United Kingdom intelligence agency Government Communications Headquarters, described an equivalent system in an internal document in 1973, but given the relatively expensive computers needed to implement it at the time, it was mostly considered a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification, and Rivest, Shamir, and Adleman devised RSA independently of Cocks' work.

Massachusetts Institute of Technology was granted for a "Cryptographic communications system and method" that used the algorithm in 1983. The patent expired on 21 September 2000. Since a paper describing the algorithm had been published in August 1977, prior to the December 1977 filing date of the patent application, regulations in much of the rest of the world precluded patents elsewhere and only the United States patent was granted. Had Cocks' work been publicly known, a patent in the US would not have been possible either.

Operation RSA involves a public Key (cryptography) and a private key. The public key can be known to everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted using the private key. The keys for the RSA algorithm are generated the following way:
Choose two distinct large random prime numbers p \, and q \,
Compute n = p q \,

n\, is used as the Modular arithmetic for both the public and private keys

Compute the totient: \phi(n) = (p-1)(q-1) \,.
Choose an integer e\, such that 1 < e\, < \phi(n)\,, and e\, and \phi (n)\, share no factors other than 1 (coprime)

e\, is released as the public key exponent

Compute d\, to satisfy the Modular arithmetic#The congruence relation d e \equiv 1\pmod{\phi(n)}\,; i.e. de = 1 + k\phi(n)\, for some integer k\,.

d\, is kept as the private key exponent

Step 1: Numbers can be Primality test#Probabilistic_tests for primality.
Step 3: changed in PKCS#1 v2.0 to \lambda(n) = {\rm lcm}(p-1, q-1) \, instead of \phi(n) = (p-1)(q-1) \,.
Step 4: A popular choice for the public exponents is e\, = 216 + 1 = 65537. Some applications choose smaller values such as e\, = 3, 5, or 35 instead. This is done to make encryption and signature verification faster on small devices like smart cards but small public exponents may lead to greater security risks.
Steps 4 and 5 can be performed with the extended Euclidean algorithm; see modular arithmetic.

public key

private key

For efficiency a different form of the private key can be stored:

p\, and q\,: the primes from the key generation,
d\mod (p - 1)\, and d\mod(q - 1)\,: often called dmp1 and dmq1.
q^{-1} \mod(p)\,: often called iqmp

All parts of the private key must be kept secret in this form. p\, and q\, are sensitive since they are the factors of n\,, and allow computation of d\, given e\,. If p\, and q\, are not stored in this form of the private key then they are securely deleted along with other intermediate values from key generation.
Although this form allows faster decryption and signing by using the Chinese Remainder Theorem, it is considerably less secure since it enables side channel attacks. This is a particular problem if implemented on smart cards, which benefit most from the improved efficiency. (Start with y = x^e mod n and let the card decrypt that. So it computes y^d \pmod{p} or y^d \pmod{q} whose results give some value z. Now, induce an error in one of the computations. Then \gcd(z-x,n) will reveal p or q.)

Fermat's little theorem

Choose two prime numbers
:p = 61 and q=53
Compute n = p q \,
:n=61*53=3233
Compute the totient \phi(n) = (p-1)(q-1) \,
:\phi(n) = (61 - 1)(53 - 1) = 3120\,
Choose e>1 coprime to 3120
:e=17
Compute d\, such that d e \equiv 1\pmod{\phi(n)}\, (d is uniquely determined by e and \phi(n)\,)
:d=2753
:17 * 2753 = 46801 = 1 + 15 * 3120.

public key

private key

padding (cryptography)

When encrypting with low encryption exponents (e.g., e = 3) and small values of the m, (i.e. m

References
- Menezes, Alfred; van Oorschot, Paul C.; and Vanstone, Scott A. Handbook of Applied Cryptography. CRC Press, October 1996. ISBN 0-8493-8523-7
- R. Rivest, A. Shamir, L. Adleman. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM, Vol. 21 (2), pp.120–126. 1978. Previously released as an MIT "Technical Memo" in April 1977. Initial publication of the RSA scheme.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.7: The RSA public-key cryptosystem, pp.881–887.
External links
- PKCS #1: RSA Cryptography Standard (RSA Laboratories website)
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