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RSA cryptosystem

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The RSA (Rivest–Shamir–Adleman) cryptosystem is a public-key cryptosystem, one of the oldest widely used for secure data transmission. The initialism "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ), the British signals intelligence agency, by the English mathematician Clifford Cocks. That system was declassified in 1997.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In a public-key cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone via the public key, but can only be decrypted by someone who knows the private key.<ref name="rsa">Template:Cite journal</ref>

The security of RSA relies on the practical difficulty of factoring the product of two large prime numbers, the "factoring problem". Breaking RSA encryption is known as the RSA problem. Whether it is as difficult as the factoring problem is an open question.<ref>Template:Cite journal 2020 interview of Peter Shor.</ref> There are no published methods to defeat the system if a large enough key is used.

RSA is a relatively slow algorithm. Because of this, it is not commonly used to directly encrypt user data. More often, RSA is used to transmit shared keys for symmetric-key cryptography, which are then used for bulk encryption–decryption.

HistoryEdit

File:Adi Shamir 2009 crop.jpg
Adi Shamir, co-inventor of RSA (the others are Ron Rivest and Leonard Adleman)

The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time.<ref>Template:Cite journal</ref> Moreover, like Diffie-Hellman, RSA is based on modular exponentiation.

Ron Rivest, Adi Shamir, and Leonard Adleman at the Massachusetts Institute of Technology made several attempts over the course of a year to create a function that was hard to invert. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches, including "knapsack-based" and "permutation polynomials". For a time, they thought what they wanted to achieve was impossible due to contradictory requirements.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In April 1977, they spent Passover at the house of a student and drank a good deal of wine before returning to their homes at around midnight.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea, and he had much of the paper ready by daybreak. The algorithm is now known as RSATemplate:Snd the initials of their surnames in same order as their paper.<ref name="SIAM">Template:Cite journal</ref>

Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), described a similar system in an internal document in 1973.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> However, given the relatively expensive computers needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His ideas and concepts were not revealed until 1997 due to its top-secret classification.

Kid-RSA (KRSA) is a simplified, insecure public-key cipher published in 1997, designed for educational purposes. Kid-RSA gives insight into RSA and other public-key ciphers, analogous to simplified DES.<ref> Jim Sauerberg. "From Private to Public Key Ciphers in Three Easy Steps". </ref><ref> Margaret Cozzens and Steven J. Miller. "The Mathematics of Encryption: An Elementary Introduction". p. 180. </ref><ref> Alasdair McAndrew. "Introduction to Cryptography with Open-Source Software". p. 12. </ref><ref> Surender R. Chiluka. "Public key Cryptography". </ref><ref> Neal Koblitz. "Cryptography As a Teaching Tool". Cryptologia, Vol. 21, No. 4 (1997). </ref>

PatentEdit

A patent describing the RSA algorithm was granted to MIT on 20 September 1983: {{#if:4405829 |[{{#ifeq:|uspto|http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=%7Chttps://patents.google.com/patent/US}}{{#iferror:{{#expr:4405829 }}|4405829}} U.S. patent {{#ifeq:Template:Replace|Template:Digits|Template:Replace|4405829}}] |{{US patent|123456|link text}}}} "Cryptographic communications system and method". From DWPI's abstract of the patent:

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The system includes a communications channel coupled to at least one terminal having an encoding device and to at least one terminal having a decoding device. A message-to-be-transferred is enciphered to ciphertext at the encoding terminal by encoding the message as a number M in a predetermined set. That number is then raised to a first predetermined power (associated with the intended receiver) and finally computed. The remainder or residue, C, is... computed when the exponentiated number is divided by the product of two predetermined prime numbers (associated with the intended receiver).{{#if:|{{#if:|}}

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A detailed description of the algorithm was published in August 1977, in Scientific American's Mathematical Games column.<ref name="SIAM" /> This preceded the patent's filing date of December 1977. Consequently, the patent had no legal standing outside the United States. Had Cocks' work been publicly known, a patent in the United States would not have been legal either.

When the patent was issued, terms of patent were 17 years. The patent was about to expire on 21 September 2000, but RSA Security released the algorithm to the public domain on 6 September 2000.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

OperationEdit

The RSA algorithm involves four steps: key generation, key distribution, encryption, and decryption.

A basic principle behind RSA is the observation that it is practical to find three very large positive integers Template:Mvar, Template:Mvar, and Template:Mvar, such that for all integers Template:Mvar (Template:Math), both <math>(m^e)^d</math> and <math>m</math> have the same remainder when divided by <math>n</math> (they are congruent modulo <math>n</math>):<math display="block">(m^e)^d \equiv m \pmod{n}.</math>However, when given only Template:Mvar and Template:Mvar, it is extremely difficult to find Template:Mvar.

The integers Template:Mvar and Template:Mvar comprise the public key, Template:Mvar represents the private key, and Template:Mvar represents the message. The modular exponentiation to Template:Mvar and Template:Mvar corresponds to encryption and decryption, respectively.

In addition, because the two exponents can be swapped, the private and public key can also be swapped, allowing for message signing and verification using the same algorithm.

Key generationEdit

The keys for the RSA algorithm are generated in the following way:

  1. Choose two large prime numbers Template:Mvar and Template:Mvar.
  2. Compute Template:Math.
  3. Compute Template:Math, where Template:Mvar is Carmichael's totient function. Since Template:Math, and since Template:Mvar and Template:Mvar are prime, Template:Math, and likewise Template:Math. Hence Template:Math.
  4. Choose an integer Template:Mvar such that Template:Math and Template:Math; that is, Template:Mvar and Template:Math are coprime.

Template:Cite journal</ref>

  2. Determine Template:Mvar as Template:Math; that is, Template:Mvar is the modular multiplicative inverse of Template:Mvar modulo Template:Math.

The public key consists of the modulus Template:Mvar and the public (or encryption) exponent Template:Mvar. The private key consists of the private (or decryption) exponent Template:Mvar, which must be kept secret. Template:Mvar, Template:Mvar, and Template:Math must also be kept secret because they can be used to calculate Template:Mvar. In fact, they can all be discarded after Template:Mvar has been computed.<ref>Applied Cryptography, John Wiley & Sons, New York, 1996. Bruce Schneier, p. 467.</ref>

Template:AnchorIn the original RSA paper,<ref name=rsa /> the Euler totient function Template:Math is used instead of Template:Math for calculating the private exponent Template:Mvar. Since Template:Math is always divisible by Template:Math, the algorithm works as well. The possibility of using Euler totient function results also from Lagrange's theorem applied to the multiplicative group of integers modulo pq. Thus any Template:Mvar satisfying Template:Math also satisfies Template:Math. However, computing Template:Mvar modulo Template:Math will sometimes yield a result that is larger than necessary (i.e. Template:Math). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent Template:Mvar at all, rather than using the optimized decryption method based on the Chinese remainder theorem described below), but some standards such as FIPS 186-4 (Section B.3.1) may require that Template:Math. Any "oversized" private exponents not meeting this criterion may always be reduced modulo Template:Math to obtain a smaller equivalent exponent.

Template:AnchorSince any common factors of Template:Math and Template:Math are present in the factorisation of Template:Math = Template:Math = Template:Math,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Self published inline it is recommended that Template:Math and Template:Math have only very small common factors, if any, besides the necessary 2.<ref name="rsa" /><ref>A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, New York, 1987. Neal Koblitz, Second edition, 1994. p. 94.</ref><ref>Template:Cite mailing list</ref>Template:Failed verification<ref>Template:Cite mailing list</ref>Template:Failed verification

Note: The authors of the original RSA paper carry out the key generation by choosing Template:Mvar and then computing Template:Mvar as the modular multiplicative inverse of Template:Mvar modulo Template:Math, whereas most current implementations of RSA, such as those following PKCS#1, do the reverse (choose Template:Mvar and compute Template:Mvar). Since the chosen key can be small, whereas the computed key normally is not, the RSA paper's algorithm optimizes decryption compared to encryption, while the modern algorithm optimizes encryption instead.<ref name="rsa" /><ref>Template:Cite IETF</ref>

Key distributionEdit

Suppose that Bob wants to send information to Alice. If they decide to use RSA, Bob must know Alice's public key to encrypt the message, and Alice must use her private key to decrypt the message.

To enable Bob to send his encrypted messages, Alice transmits her public key Template:Math to Bob via a reliable, but not necessarily secret, route. Alice's private key Template:Math is never distributed.

EncryptionEdit

After Bob obtains Alice's public key, he can send a message Template:Mvar to Alice.

To do it, he first turns Template:Mvar (strictly speaking, the un-padded plaintext) into an integer Template:Mvar (strictly speaking, the padded plaintext), such that Template:Math by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext Template:Mvar, using Alice's public key Template:Mvar, corresponding to

<math display="block">c \equiv m^e \pmod{n}.</math>

This can be done reasonably quickly, even for very large numbers, using modular exponentiation. Bob then transmits Template:Mvar to Alice. Note that at least nine values of Template:Mvar will yield a ciphertext Template:Mvar equal to Template:Mvar,Template:Efn but this is very unlikely to occur in practice.

DecryptionEdit

Alice can recover Template:Mvar from Template:Mvar by using her private key exponent Template:Mvar by computing

<math display="block">c^d \equiv (m^e)^d \equiv m \pmod{n}.</math>

Given Template:Mvar, she can recover the original message Template:Mvar by reversing the padding scheme.

ExampleEdit

Here is an example of RSA encryption and decryption:Template:Efn

  1. Choose two distinct prime numbers, such as
    <math>p = 61</math> and <math>q = 53</math>.
  2. Compute Template:Math giving
    <math>n = 61\times 53 = 3233.</math>
  3. Compute the Carmichael's totient function of the product as Template:Math giving
    <math>\lambda(3233) = \operatorname{lcm}(60, 52) = 780.</math>
  4. Choose any number Template:Math that is coprime to 780. Choosing a prime number for Template:Mvar leaves us only to check that Template:Mvar is not a divisor of 780.
    Let <math>e = 17</math>.
  5. Compute Template:Mvar, the modular multiplicative inverse of Template:Math, yielding
    <math display="block">d = 413,</math> as <math>1 = (17 \times 413) \bmod 780.</math>

The public key is Template:Math. For a padded plaintext message Template:Mvar, the encryption function is <math display="block">\begin{align} 

c(m) &= m^{e} \bmod n \\
     &= m^{17} \bmod 3233.

\end{align}</math>

The private key is Template:Math. For an encrypted ciphertext Template:Mvar, the decryption function is <math display="block">\begin{align} 

m(c) &= c^{d} \bmod n \\
     &= c^{413} \bmod 3233.

\end{align}</math>

For instance, in order to encrypt Template:Math, one calculates <math display="block">c = 65^{17} \bmod 3233 = 2790.</math>

To decrypt Template:Math, one calculates <math display="block">m = 2790^{413} \bmod 3233 = 65.</math>

Both of these calculations can be computed efficiently using the square-and-multiply algorithm for modular exponentiation. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor Template:Math (obtained from the freely available public key) back to the primes Template:Mvar and Template:Mvar. Template:Mvar, also from the public key, is then inverted to get Template:Mvar, thus acquiring the private key.

Practical implementations use the Chinese remainder theorem to speed up the calculation using modulus of factors (mod pq using mod p and mod q).

The values Template:Mvar, Template:Mvar and Template:Mvar, which are part of the private key are computed as follows: <math display="block">\begin{align} 

          d_p &= d \bmod (p-1) = 413 \bmod (61 - 1) = 53, \\
          d_q &= d \bmod (q-1) = 413 \bmod (53 - 1) = 49, \\
 q_\text{inv} &= q^{-1} \bmod p = 53^{-1} \bmod 61 = 38 \\
              &\Rightarrow (q_\text{inv} \times q) \bmod p = 38 \times 53 \bmod 61 = 1.

\end{align}</math>

Here is how Template:Mvar, Template:Mvar and Template:Mvar are used for efficient decryption (encryption is efficient by choice of a suitable Template:Mvar and Template:Mvar pair): <math display="block">\begin{align} 

 m_1 &= c^{d_p} \bmod p = 2790^{53} \bmod 61 = 4, \\
 m_2 &= c^{d_q} \bmod q = 2790^{49} \bmod 53 = 12, \\
   h &= (q_\text{inv} \times (m_1 - m_2)) \bmod p = (38 \times -8) \bmod 61 = 1, \\
   m &= m_2 + h \times q = 12 + 1 \times 53 = 65.

\end{align}</math>

Signing messagesEdit

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 from Alice, since anyone can use Bob's public key to send him encrypted messages. In order to verify the origin of a message, RSA can also be used to sign a message.

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 Template:Mvar (modulo Template:Mvar) (as she does when decrypting a message), and attaches it as a "signature" to the message. 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 Template:Mvar (modulo Template:Mvar) (as he does when encrypting a message), and compares the resulting hash value with the message's hash value. If the two agree, he knows that the author of the message was in possession of Alice's private key and that the message has not been tampered with since being sent.

This works because of exponentiation rules: <math display="block">h = \operatorname{hash}(m),</math> <math display="block">(h^e)^d = h^{ed} = h^{de} = (h^d)^e \equiv h \pmod{n}.</math>

Thus the keys may be swapped without loss of generality, that is, a private key of a key pair may be used either to:

  1. Decrypt a message only intended for the recipient, which may be encrypted by anyone having the public key (asymmetric encrypted transport).
  2. Encrypt a message which may be decrypted by anyone, but which can only be encrypted by one person; this provides a digital signature.

Proofs of correctnessEdit

Proof using Fermat's little theoremEdit

The proof of the correctness of RSA is based on Fermat's little theorem, stating that Template:Math for any integer Template:Mvar and prime Template:Mvar, not dividing Template:Mvar.Template:Refn

We want to show that <math display="block">(m^e)^d \equiv m \pmod{pq}</math> for every integer Template:Mvar when Template:Mvar and Template:Mvar are distinct prime numbers and Template:Mvar and Template:Mvar are positive integers satisfying Template:Math.

Since Template:Math is, by construction, divisible by both Template:Math and Template:Math, we can write <math display="block">ed - 1 = h(p - 1) = k(q - 1)</math> for some nonnegative integers Template:Mvar and Template:Mvar.Template:Refn

To check whether two numbers, such as Template:Mvar and Template:Mvar, are congruent Template:Math, it suffices (and in fact is equivalent) to check that they are congruent Template:Math and Template:Math separately.Template:Refn

To show Template:Math, we consider two cases:

  1. If Template:Math, Template:Mvar is a multiple of Template:Mvar. Thus med is a multiple of Template:Mvar. So Template:Math.
  2. If Template:Math,
    <math>m^{ed} = m^{ed - 1} m = m^{h(p - 1)} m = (m^{p - 1})^h m \equiv 1^h m \equiv m \pmod{p},</math>
    where we used Fermat's little theorem to replace Template:Math with 1.

The verification that Template:Math proceeds in a completely analogous way:

  1. If Template:Math, med is a multiple of Template:Mvar. So Template:Math.
  2. If Template:Math,
    <math>m^{ed} = m^{ed - 1} m = m^{k(q - 1)} m = (m^{q - 1})^k m \equiv 1^k m \equiv m \pmod{q}.</math>

This completes the proof that, for any integer Template:Mvar, and integers Template:Mvar, Template:Mvar such that Template:Math, <math display="block">(m^e)^d \equiv m \pmod{pq}.</math>

NotesEdit

Template:Reflist

Proof using Euler's theoremEdit

Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on Euler's theorem.

We want to show that Template:Math, where Template:Math is a product of two different prime numbers, and Template:Mvar and Template:Mvar are positive integers satisfying Template:Math. Since Template:Mvar and Template:Mvar are positive, we can write Template:Math for some non-negative integer Template:Mvar. Assuming that Template:Mvar is relatively prime to Template:Mvar, we have <math display="block">m^{ed} = m^{1 + h\varphi(n)} = m (m^{\varphi(n)})^h \equiv m (1)^h \equiv m \pmod{n},</math>

where the second-last congruence follows from Euler's theorem.

More generally, for any Template:Mvar and Template:Mvar satisfying Template:Math, the same conclusion follows from Carmichael's generalization of Euler's theorem, which states that Template:Math for all Template:Mvar relatively prime to Template:Mvar.

When Template:Mvar is not relatively prime to Template:Mvar, the argument just given is invalid. This is highly improbable (only a proportion of Template:Math numbers have this property), but even in this case, the desired congruence is still true. Either Template:Math or Template:Math, and these cases can be treated using the previous proof.

PaddingEdit

Attacks against plain RSAEdit

There are a number of attacks against plain RSA as described below.

Padding schemesEdit

To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value Template:Mvar before encrypting it. This padding ensures that Template:Mvar 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. Because these schemes pad the plaintext Template:Mvar with some number of additional bits, the size of the un-padded message Template:Mvar must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks that may be facilitated by a predictable message structure. Early versions of the PKCS#1 standard (up to version 1.5) used a construction that appears to make RSA semantically secure. However, at Crypto 1998, Bleichenbacher showed that this version is vulnerable to a practical adaptive chosen-ciphertext attack. Furthermore, at Eurocrypt 2000, Coron et al.<ref>Template:Cite book</ref> showed that for some types of messages, this padding does not provide a high enough level of security. Later versions of the standard include Optimal Asymmetric Encryption Padding (OAEP), which prevents these attacks. As such, OAEP should be used in any new application, and PKCS#1 v1.5 padding should be replaced wherever possible. 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).

Secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption. Two USA patents on PSS were granted ({{#if:6266771 |[{{#ifeq:|uspto|http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=%7Chttps://patents.google.com/patent/US}}{{#iferror:{{#expr:6266771 }}|6266771}} U.S. patent {{#ifeq:Template:Replace|Template:Digits|Template:Replace|6266771}}] |{{US patent|123456|link text}}}} and {{#if:7036014 |[{{#ifeq:|uspto|http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=%7Chttps://patents.google.com/patent/US}}{{#iferror:{{#expr:7036014 }}|7036014}} U.S. patent {{#ifeq:Template:Replace|Template:Digits|Template:Replace|7036014}}] |{{US patent|123456|link text}}}}); however, these patents expired on 24 July 2009 and 25 April 2010 respectively. Use of PSS no longer seems to be encumbered by patents.Template:Original research inline Note that using different RSA key pairs for encryption and signing is potentially more secure.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Security and practical considerationsEdit

Using the Chinese remainder algorithmEdit

For efficiency, many popular crypto libraries (such as OpenSSL, Java and .NET) use for decryption and signing the following optimization based on the Chinese remainder theorem.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Fact The following values are precomputed and stored as part of the private key:

  • <math>p</math> and <math>q</math>Template:Snd the primes from the key generation,
  • <math>d_P = d \pmod{p - 1},</math>
  • <math>d_Q = d \pmod{q - 1},</math>
  • <math>q_\text{inv} = q^{-1} \pmod{p}.</math>

These values allow the recipient to compute the exponentiation Template:Math more efficiently as follows: Template:Indent<math>m_1 = c^{d_P} \pmod{p}</math>, Template:Indent<math>m_2 = c^{d_Q} \pmod{q}</math>, Template:Indent<math>h = q_\text{inv}(m_1 - m_2) \pmod{p}</math>,Template:Efn Template:Indent<math>m = m_2 + hq</math>.

This is more efficient than computing exponentiation by squaring, even though two modular exponentiations have to be computed. The reason is that these two modular exponentiations both use a smaller exponent and a smaller modulus.

Integer factorization and the RSA problemEdit

Template:See also The security of the RSA cryptosystem is based on two mathematical problems: the problem of factoring large numbers and the RSA problem. 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.<ref>Template:Cite book</ref>

The RSA problem is defined as the task of taking Template:Mvarth roots modulo a composite Template:Mvar: recovering a value Template:Mvar such that Template:Math, where Template:Math is an RSA public key, and Template:Mvar is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus Template:Mvar. With the ability to recover prime factors, an attacker can compute the secret exponent Template:Mvar from a public key Template:Math, then decrypt Template:Mvar using the standard procedure. To accomplish this, an attacker factors Template:Mvar into Template:Mvar and Template:Mvar, and computes Template:Math that allows the determination of Template:Mvar from Template:Mvar. 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.

The first RSA-512 factorization in 1999 used hundreds of computers and required the equivalent of 8,400 MIPS years, over an elapsed time of about seven months.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> By 2009, Benjamin Moody could factor an 512-bit RSA key in 73 days using only public software (GGNFS) and his desktop computer (a dual-core Athlon64 with a 1,900 MHz CPU). Just less than 5 gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process.

Rivest, Shamir, and Adleman noted<ref name="rsa" /> that Miller has shown that – assuming the truth of the extended Riemann hypothesis – finding Template:Mvar from Template:Mvar and Template:Mvar is as hard as factoring Template:Mvar into Template:Mvar and Template:Mvar (up to a polynomial time difference).<ref>Template:Cite conference</ref> However, Rivest, Shamir, and Adleman noted, in section IX/D of their paper, that they had not found a proof that inverting RSA is as hard as factoring.

Template:As of, the largest publicly known factored RSA number had 829 bits (250 decimal digits, RSA-250).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Its factorization, by a state-of-the-art distributed implementation, took about 2,700 CPU-years. In practice, RSA keys are typically 1024 to 4096 bits long. In 2003, RSA Security estimated that 1024-bit keys were likely to become crackable by 2010.<ref name="twirl">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> As of 2020, it is not known whether such keys can be cracked, but minimum recommendations have moved to at least 2048 bits.<ref name="keymanagement">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> It is generally presumed that RSA is secure if Template:Mvar is sufficiently large, outside of quantum computing.

If Template:Mvar is 300 bits or shorter, it can be factored in a few hours on a personal computer, using software already freely available. Keys of 512 bits have been shown to be practically breakable in 1999, when RSA-155 was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit code-signing certificates that may have been factored were reported in 2011.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> A theoretical hardware device named TWIRL, described by Shamir and Tromer in 2003, called into question the security of 1024-bit keys.<ref name="twirl" />

In 1994, Peter Shor showed that a quantum computer – if one could ever be practically created for the purpose – would be able to factor in polynomial time, breaking RSA; see Shor's algorithm.

Faulty key generationEdit

Template:More citations needed Template:See also

Finding the large primes Template:Mvar and Template:Mvar is usually done by testing random numbers of the correct size with probabilistic primality tests that quickly eliminate virtually all of the nonprimes.

The numbers Template:Mvar and Template:Mvar should not be "too close", lest the Fermat factorization for Template:Mvar be successful. If Template:Math is less than Template:Math (Template:Math, which even for "small" 1024-bit values of Template:Mvar is Template:Val), solving for Template:Mvar and Template:Mvar is trivial. Furthermore, if either Template:Math or Template:Math has only small prime factors, Template:Mvar can be factored quickly by Pollard's p − 1 algorithm, and hence such values of Template:Mvar or Template:Mvar should be discarded.

It is important that the private exponent Template:Mvar be large enough. Michael J. Wiener showed that if Template:Mvar is between Template:Mvar and Template:Math (which is quite typical) and Template:Math, then Template:Mvar can be computed efficiently from Template:Mvar and Template:Mvar.<ref name="wiener">Template:Cite journal</ref>

There is no known attack against small public exponents such as Template:Math, provided that the proper padding is used. Coppersmith's attack has many applications in attacking RSA specifically if the public exponent Template:Mvar is small and if the encrypted message is short and not padded. 65537 is a commonly used value for Template:Mvar; 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 Template:Mvar smaller than 65537, but does not state a reason for this restriction.

In October 2017, a team of researchers from Masaryk University announced the ROCA vulnerability, which affects RSA keys generated by an algorithm embodied in a library from Infineon known as RSALib. A large number of smart cards and trusted platform modules (TPM) were shown to be affected. Vulnerable RSA keys are easily identified using a test program the team released.<ref name=nemecsys>Template:Cite conference</ref>

Importance of strong random number generationEdit

A cryptographically strong random number generator, which has been properly seeded with adequate entropy, must be used to generate the primes Template:Mvar and Template:Mvar. An analysis comparing millions of public keys gathered from the Internet was carried out in early 2012 by Arjen K. Lenstra, James P. Hughes, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.2% of the keys using only Euclid's algorithm.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>Template:Self published inline

They exploited a weakness unique to cryptosystems based on integer factorization. If Template:Math is one public key, and Template:Math is another, then if by chance Template:Math (but Template:Mvar is not equal to Template:Mvar'), then a simple computation of Template:Math factors both Template:Mvar and Template:Mvar', totally compromising both keys. Lenstra et al. note that this problem can be minimized by using a strong random seed of bit length twice the intended security level, or by employing a deterministic function to choose Template:Mvar given Template:Mvar, instead of choosing Template:Mvar and Template:Mvar independently.

Nadia Heninger was part of a group that did a similar experiment. They used an idea of Daniel J. Bernstein to compute the GCD of each RSA key Template:Mvar against the product of all the other keys Template:Mvar' they had found (a 729-million-digit number), instead of computing each Template:Math separately, thereby achieving a very significant speedup, since after one large division, the GCD problem is of normal size.

Heninger says in her blog that the bad keys occurred almost entirely in embedded applications, including "firewalls, routers, VPN devices, remote server administration devices, printers, projectors, and VOIP phones" from more than 30 manufacturers. Heninger explains that the one-shared-prime problem uncovered by the two groups results from situations where the pseudorandom number generator is poorly seeded initially, and then is reseeded between the generation of the first and second primes. Using seeds of sufficiently high entropy obtained from key stroke timings or electronic diode noise or atmospheric noise from a radio receiver tuned between stations should solve the problem.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Strong random number generation is important throughout every phase of public-key cryptography. For instance, if a weak generator is used for the symmetric keys that are being distributed by RSA, then an eavesdropper could bypass RSA and guess the symmetric keys directly.

Timing attacksEdit

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, Eve can deduce the decryption key Template:Mvar quickly. 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.g., from a Secure Sockets Layer (SSL)-enabled webserver).<ref name="Boneh03">Template:Cite conference</ref> This attack takes advantage of information leaked by the Chinese remainder theorem optimization used by many RSA implementations.

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. RSA blinding makes use of the multiplicative property of RSA. Instead of computing Template:Math, Alice first chooses a secret random value Template:Mvar and computes Template:Math. The result of this computation, after applying Euler's theorem, is Template:Math, and so the effect of Template:Mvar can be removed by multiplying by its inverse. A new value of Template:Mvar 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 attacksEdit

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). Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the Secure Sockets Layer protocol and to recover session keys. 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.

A variant of this attack, dubbed "BERserk", came back in 2014.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> It impacted the Mozilla NSS Crypto Library, which was used notably by Firefox and Chrome.

Side-channel analysis attacksEdit

A side-channel attack using branch-prediction analysis (BPA) has been described. 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. Often these processors also implement simultaneous multithreading (SMT). 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",<ref>Template:Cite conference</ref> 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.

A power-fault attack on RSA implementations was described in 2010.<ref>Template:Cite book</ref> The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.

Tricky implementationEdit

There are many details to keep in mind in order to implement RSA securely (strong PRNG, acceptable public exponent, etc.). This makes the implementation challenging, to the point that the book Practical Cryptography With Go suggests avoiding RSA if possible.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

ImplementationsEdit

Some cryptography libraries that provide support for RSA include:

See alsoEdit

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NotesEdit

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ReferencesEdit

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Further readingEdit

External linksEdit

  • The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977, {{#if:4405829

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