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This article is about an algorithm for public-key encryption. For the Republic of South Africa, see South Africa. The Republic of South Africa (also known by other official names) is a country located at the southern tip of the continent of Africa For other uses, see RSA (disambiguation).

In cryptography, RSA is an algorithm for public-key cryptography. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key It was the first algorithm known to be suitable for signing as well as encryption, and one of the first great advances in public key cryptography. A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a handwritten Signature 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. Electronic commerce, commonly known as e-commerce' or eCommerce, consists of the buying and selling of products or services over electronic

Contents

History

The algorithm was publicly described in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT; the letters RSA are the initials of their surnames. Ronald Linn Rivest (born 1947, Schenectady, New York) is a cryptographer. Adi Shamir (עדי שמיר born 1952 is an Israeli cryptographer. Leonard Max Adleman (born December 31, 1945) is a theoretical computer scientist and professor of Computer science and Molecular biology [1]

Clifford Cocks, a British mathematician working for the UK intelligence agency GCHQ, 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. Clifford Christopher Cocks, CB, (born 1951 is a British Mathematician and Cryptographer at GCHQ who invented the widely-used encryption algorithm A mathematician is a person whose primary area of study and research is the field of Mathematics. The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located The Government Communications Headquarters (GCHQ is a British Intelligence agency responsible for providing Signals intelligence (SIGINT and Information 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.

MIT was granted US patent 4405829 for a "Cryptographic communications system and method" that used the algorithm in 1983. The patent expired on 21 September 2000. Events 1217 - The Estonian tribal leader Lembitu of Lehola was killed in a battle against Teutonic Knights. 2000 ( MM) was a Leap year that started on Saturday of the Common Era, in accordance with the Gregorian calendar. Since a paper describing the algorithm had been published in August 1977,[1] 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 US patent was granted. The filing date of a Patent application is the date the patent application was filed in one or more Patent offices In the United States, if a patent A patent application is a request pending at a Patent office for the grant of a Patent for the Invention described and claimed by that application A patent is a set of Exclusive rights granted by a State to an inventor or his assignee for a fixed period of time in exchange for a disclosure of an The United States of America —commonly referred to as the Had Cocks' work been publicly known, a patent in the US might not have been possible either.

Operation

RSA involves a public key and a private key. In Cryptography, a key is a piece of information (a Parameter) that determines the functional output of a cryptographic algorithm 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:

  1. Choose two distinct large random prime numbers p and q
  2. Compute n = pq\,
    • n\, is used as the modulus for both the public and private keys
  3. Compute the totient: \varphi(n) = (p-1)(q-1) \,. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to
  4. Choose an integer e such that 1 < e < \varphi(n), and e\, and \varphi (n) share no factors other than 1 (i. e. e and \varphi (n) are coprime)
    • e is released as the public key exponent
  5. Compute d to satisfy the congruence relation d e \equiv 1\pmod{\varphi(n)}; i. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers e. de = 1 + k\varphi(n) for some integer k.
    • d is kept as the private key exponent

Notes on the above steps:


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.

Messages

Encryption

Alice transmits her public key (n,e)\, to Bob and keeps the private key secret. The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as Cryptography and Physics. The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as Cryptography and Physics. 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 scheme. 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. Exponentiating by squaring is an Algorithm used for the fast computation of large Integer powers of a Number. Bob then transmits c\, to Alice.

Decryption

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}\,. Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a Prime number, then for any Integer a

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, they are relatively prime to each other, so the fact that both primes divide med − m implies their product pq\, divides med − m, which means

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

Thus,

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

A worked example

Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but you can also use OpenSSL to generate and examine a real keypair.

  1. Choose two prime numbers
    p = 61 and q = 53
  2. Compute n = p q \,
    n = 61 * 53 = 3233
  3. Compute the totient \varphi(n) = (p-1)(q-1) \,
    \varphi(n) = (61 - 1)(53 - 1) = 3120\,
  4. Choose e > 1 coprime to 3120
    e = 17
  5. Compute d\, such that d e \equiv 1\pmod{\varphi(n)}\, e. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to g. , by computing the modular multiplicative inverse of e modulo \varphi(n)\,:
    d = 2753
    17 * 2753 = 46801 = 1 + 15 * 3120. The modular multiplicative inverse of an Integer n modulo p is an integer m such that n -1 ≡


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. Exponentiating by squaring is an Algorithm used for the fast computation of large Integer powers of a Number. Modular exponentiation is a type of Exponentiation performed over a modulus.

Padding schemes

When used in practice, RSA is generally combined with some padding scheme. In Cryptography, padding refers to a number of distinct practices The goal of the padding scheme is to prevent a number of attacks that potentially work against RSA without padding:

To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. In Cryptography, padding refers to a number of distinct practices This padding ensures that m does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts.

Standards such as PKCS#1 have been carefully designed to securely pad messages prior to RSA encryption. In Cryptography, PKCS#1 is the first of a family of standards called Public-Key Cryptography Standards (PKCS, published by RSA Laboratories. Because these schemes pad the plaintext m with some number of additional bits, the size of the un-padded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks which may be facilitated by a predictable message structure. Early versions of the PKCS#1 standard (up to version 1. 5) used a construction that turned RSA into a semantically secure encryption scheme. This version was later found vulnerable to a practical adaptive chosen ciphertext attack. An adaptive chosen-ciphertext attack (abbreviated as CCA2) is an interactive form of Chosen-ciphertext attack in which an attacker sends a number of ciphertexts to Later versions of the standard include Optimal Asymmetric Encryption Padding (OAEP), which prevents these attacks. This article is about the padding scheme used in public-key cryptography The PKCS#1 standard also incorporates processing schemes designed to provide additional security for RSA signatures, e. g. , the Probabilistic Signature Scheme for RSA (RSA-PSS). In Cryptography, PKCS#1 is the first of a family of standards called Public-Key Cryptography Standards (PKCS, published by RSA Laboratories.

Signing messages

Suppose Alice uses Bob's public key to send him an encrypted message. In the message, she can claim to be Alice but Bob has no way of verifying that the message was actually from Alice since anyone can use Bob's public key to send him encrypted messages. So, in order to verify the origin of a message, RSA can also be used to sign a message. A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a handwritten Signature

Suppose Alice wishes to send a signed message to Bob. She can use her own private key to do so. She produces a hash value of the message, raises it to the power of d mod n (as she does when decrypting a message), and attaches it as a "signature" to the message. A cryptographic Hash function is a transformation that takes an input (or 'message' and returns a fixed-size string which is called the hash value (sometimes When Bob receives the signed message, he uses the same hash algorithm in conjunction with Alice's public key. He raises the signature to the power of e mod n (as he does when encrypting a message), and compares the resulting hash value with the message's actual hash value. If the two agree, he knows that the author of the message was in possession of Alice's secret key, and that the message has not been tampered with since.

Note that secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption, and that the same key should never be used for both encryption and signing purposes. In Cryptography, PKCS#1 is the first of a family of standards called Public-Key Cryptography Standards (PKCS, published by RSA Laboratories.

Security

See also: RSA Factoring Challenge and Integer factorization records

The security of the RSA cryptosystem is based on two mathematical problems: the problem of factoring large numbers and the RSA problem. The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18 1991 to encourage research into Computational number theory Numbers of a general form The first very large distributed factorisation was RSA129 a challenge number described in the Scientific American article of 1977 which first popularised In Cryptography, the RSA problem is the task of finding e th roots modulo a Composite number N whose factors Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are hard, i. e. , no efficient algorithm exists for solving them. Providing security against partial decryption may require the addition of a secure padding scheme. In Cryptography, padding refers to a number of distinct practices

The RSA problem is defined as the task of taking eth roots modulo a composite n: recovering a value m such that c=me mod n, where (n, e) is an RSA public key and c is an RSA ciphertext. In Cryptography, the RSA problem is the task of finding e th roots modulo a Composite number N whose factors Currently the most promising approach to solving the RSA problem is to factor the modulus n. With the ability to recover prime factors, an attacker can compute the secret exponent d from a public key (n, e), then decrypt c using the standard procedure. To accomplish this, an attacker factors n into p and q, and computes (p-1)(q-1) which allows the determination of d from e. No polynomial-time method for factoring large integers on a classical computer has yet been found, but it has not been proven that none exists. See integer factorization for a discussion of this problem.

As of 2005, the largest number factored by a general-purpose factoring algorithm was 663 bits long (see RSA-200), using a state-of-the-art distributed implementation. Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. In Mathematics, the RSA numbers are a set of large Semiprimes (numbers with exactly two Prime factors that are part of the RSA Factoring Challenge RSA keys are typically 1024–2048 bits long. Some experts believe that 1024-bit keys may become breakable in the near term (though this is disputed); few see any way that 4096-bit keys could be broken in the foreseeable future. Therefore, it is generally presumed that RSA is secure if n is sufficiently large. If n is 256 bits or shorter, it can be factored in a few hours on a personal computer, using software already freely available. A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication A personal computer ( PC) is any Computer whose original sales price size and capabilities make it useful for individuals and which is intended to be operated Keys of 512 bits (or less) have been shown to be practically breakable in 1999 when RSA-155 was factored by using several hundred computers. Year 1999 ( MCMXCIX) was a Common year starting on Friday (link will display full 1999 Gregorian calendar) In Mathematics, the RSA numbers are a set of large Semiprimes (numbers with exactly two Prime factors that are part of the RSA Factoring Challenge A theoretical hardware device named TWIRL and described by Shamir and Tromer in 2003 called into question the security of 1024 bit keys. In Cryptography and Number theory, TWIRL ( The Weizmann Institute Relation Locator) is a hypothetical hardware device designed to speed up It is currently recommended that n be at least 2048 bits long.

In 1994, Peter Shor published Shor's algorithm, showing that a quantum computer could in principle perform the factorization in polynomial time. Peter Williston Shor (born August 14, 1959) is an American theoretical computer scientist most famous for his work on Quantum computation Shor's algorithm, first introduced by mathematician Peter Shor, is a quantum Algorithm for Integer factorization. A quantum computer is a device for Computation that makes direct use of distinctively Quantum mechanical Phenomena, such as superposition However, quantum computation is still in the early stages of development and may never prove to be practical.

Practical considerations

Key generation

Finding the large primes p and q is usually done by testing random numbers of the right size with probabilistic primality tests which quickly eliminate virtually all non-primes. A primality test is an Algorithm for determining whether an input number is prime.

p and q should not be 'too close', lest the Fermat factorization for n be successful, if p-q, for instance is less than 2n1/4 (which for even small 1024-bit values of n is 3x1077) solving for p and q is trivial. Fermat 's factorization method is based on the representation of an odd Integer as the Difference of two squares: N = a^2 - Furthermore, if either p-1 or q-1 has only small prime factors, n can be factored quickly by Pollard's p − 1 algorithm, and these values of p or q should therefore be discarded as well. Pollard's p &minus 1 algorithm is a number theoretic Integer factorization Algorithm, invented by John Pollard in 1974

It is important that the secret key d be large enough. Michael J. Wiener showed[5] that if p is between q and 2q (which is quite typical) and d < n1/4/3, then d can be computed efficiently from n and e. There is no known attack against small public exponents such as e=3, provided that proper padding is used. However, when no padding is used or when the padding is improperly implemented then small public exponents have a greater risk of leading to an attack, such as for example the unpadded plaintext vulnerability listed above. 65537 is a commonly used value for e. This value can be regarded as a compromise between avoiding potential small exponent attacks and still allowing efficient encryptions (or signature verification). The NIST Special Publication on Computer Security (SP 800-78 Rev 1 of August 2007) does not allow public exponents e smaller than 65537, but does not state a reason for this restriction.

Speed

RSA is much slower than DES and other symmetric cryptosystems. The Data Encryption Standard ( DES) is a Cipher (a method for Encrypting information selected by NBS as an official Federal Information Symmetric-key algorithms are a class of Algorithms for Cryptography that use trivially related often identical Cryptographic keys for both decryption In practice, Bob typically encrypts a secret message with a symmetric algorithm, encrypts the (comparatively short) symmetric key with RSA, and transmits both the RSA-encrypted symmetric key and the symmetrically-encrypted message to Alice. The names Alice and Bob are commonly used placeholders for archetypal characters in fields such as Cryptography and Physics.

This procedure raises additional security issues. For instance, it is of utmost importance to use a strong random number generator for the symmetric key, because otherwise Eve (an eavesdropper wanting to see what was sent) could bypass RSA by guessing the symmetric key. A random number generator (often abbreviated as RNG is a computational or physical device designed to generate a sequence of Numbers or symbols that lack any

Key distribution

As with all ciphers, how RSA public keys are distributed is important to security. Key distribution must be secured against a man-in-the-middle attack. In Cryptography, the man-in-the-middle attack or bucket-brigade attack (often abbreviated MITM) sometimes Janus attack, is a Suppose Eve has some way to give Bob arbitrary keys and make him believe they belong to Alice. Suppose further that Eve can intercept transmissions between Alice and Bob. Eve sends Bob her own public key, which Bob believes to be Alice's. Eve can then intercept any ciphertext sent by Bob, decrypt it with her own secret key, keep a copy of the message, encrypt the message with Alice's public key, and send the new ciphertext to Alice. In principle, neither Alice nor Bob would be able to detect Eve's presence. Defenses against such attacks are often based on digital certificates or other components of a public key infrastructure. In Cryptography, a public key certificate (or identity certificate) is an electronic document which incorporates a Digital signature to bind together a In Cryptography, a public key infrastructure ( PKI) is an arrangement that binds Public keys with respective user identities by means of a Certificate

Timing attacks

Kocher described a new attack on RSA in 1995: if the attacker Eve knows Alice's hardware in sufficient detail and is able to measure the decryption times for several known ciphertexts, she can deduce the decryption key d quickly. Paul Carl Kocher (born June 11, 1973) is an American Cryptographer and Cryptography Consultant, currently the This attack can also be applied against the RSA signature scheme. In 2003, Boneh and Brumley demonstrated a more practical attack capable of recovering RSA factorizations over a network connection (e. Year 2003 ( MMIII) was a Common year starting on Wednesday of the Gregorian calendar. Dan Boneh (דן בונה ˈdæn boʊˈneɪ is a Professor of Computer Science and Electrical Engineering at Stanford University. g. , from a Secure Socket Layer (SSL)-enabled webserver). Transport Layer Security ( TLS) and its predecessor Secure Sockets Layer ( SSL) are Cryptographic protocols that provide secure This attack takes advantage of information leaked by the Chinese remainder theorem optimization used by many RSA implementations. The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra.

One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. However, this approach can significantly reduce performance. Instead, most RSA implementations use an alternate technique known as cryptographic blinding. In Cryptography, blinding is a technique by which an agent can provide a service to (i RSA blinding makes use of the multiplicative property of RSA. Instead of computing cd mod n, Alice first chooses a secret random value r and computes (rec)d mod n. The result of this computation is r m mod n and so the effect of r can be removed by multiplying by its inverse. A new value of r is chosen for each ciphertext. With blinding applied, the decryption time is no longer correlated to the value of the input ciphertext and so the timing attack fails.

Adaptive chosen ciphertext attacks

In 1998, Daniel Bleichenbacher described the first practical adaptive chosen ciphertext attack, against RSA-encrypted messages using the PKCS #1 v1 padding scheme (a padding scheme randomizes and adds structure to an RSA-encrypted message, so it is possible to determine whether a decrypted message is valid. Year 1998 ( MCMXCVIII) was a Common year starting on Thursday (link will display full 1998 Gregorian calendar) Daniel Bleichenbacher (born August 4, 1964) is a Swiss Cryptographer, currently a researcher at Bell Labs. An adaptive chosen-ciphertext attack (abbreviated as CCA2) is an interactive form of Chosen-ciphertext attack in which an attacker sends a number of ciphertexts to In Cryptography, padding refers to a number of distinct practices ) Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the Secure Socket Layer protocol, and to recover session keys. Transport Layer Security ( TLS) and its predecessor Secure Sockets Layer ( SSL) are Cryptographic protocols that provide secure As a result of this work, cryptographers now recommend the use of provably secure padding schemes such as Optimal Asymmetric Encryption Padding, and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks. This article is about the padding scheme used in public-key cryptography

Branch prediction analysis attacks

Branch prediction analysis is also called BPA. Many processors use a branch predictor to determine whether a conditional branch in the instruction flow of a program is likely to be taken or not. In Computer architecture, a branch predictor is the part of a processor that determines whether a Conditional branch in the instruction Usually these processors also implement simultaneous multithreading (SMT). Simultaneous multithreading, often abbreviated as SMT, is a technique for improving the overall efficiency of Superscalar CPUs with Hardware Branch prediction analysis attacks use a spy process to discover (statistically) the private key when processed with these processors.

Simple Branch Prediction Analysis (SBPA) claims to improve BPA in a non-statistical way. In their paper, "On the Power of Simple Branch Prediction Analysis", the authors of SBPA (Onur Aciicmez and Cetin Kaya Koc) claim to have discovered 508 out of 512 bits of an RSA key in 10 iterations.

See also

Notes

  1. ^ a b SIAM News, Volume 36, Number 5, June 2003, "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders", by Sara Robinson
  2. ^ [1] Boneh, Dan. Clifford Christopher Cocks, CB, (born 1951 is a British Mathematician and Cryptographer at GCHQ who invented the widely-used encryption algorithm Quantum cryptography, or quantum key distribution (QKD uses Quantum mechanics to guarantee secure communication In Cryptography, key size or key length is the size (usually measured in bits or bytes of the key used in a cryptographic algorithm (such as a Cipher Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources Diffie-Hellman key exchange ( D-H) is a Cryptographic protocol that allows two parties that have no prior knowledge of each other to jointly establish a shared secret The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18 1991 to encourage research into Computational number theory This is a categorized list of notable Patents and Patent applications involving Computer programs often labelled Software patents The patents are listed according Twenty Years of attacks on the RSA Cryptosystem. Notices of the American Mathematical Society (AMS), Vol. 46, No. 2, pp. 203-213, 1999
  3. ^ Johan Håstad, "On using RSA with Low Exponent in a Public Key Network", Crypto 85
  4. ^ Don Coppersmith, "Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities", Journal of Cryptology, v. 10, n. 4, Dec. 1997
  5. ^ Wiener, Michael J. (May 1990), “Cryptanalysis of short RSA secret exponents”, Information Theory, IEEE Transactions on 36 (3): 553-558, DOI 10. 1109/18. 54902 

References

External links

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Dictionary

RSA

-noun

  1. (initialism) (cryptography) the Rivest-Shamir-Adleman encryption algorithm
  2. (initialism) Royal Society of Arts
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