RSA (cryptosystem)

RSA

General
Designers	829-bit key has been broken.

RSA (Rivest–Shamir–Adleman) is a

In a public-key

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.^[1]

The security of RSA relies on the practical difficulty of

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.

History

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.^[4] 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.^[5] 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.^[6] 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 RSA – the initials of their surnames in same order as their paper.^[7]

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. Some people feel that learning Kid-RSA gives insight into RSA and other public-key ciphers, analogous to simplified DES.^{[9][10][11][12][13]}

Patent

A

's abstract of the patent:

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).

A detailed description of the algorithm was published in August 1977, in Scientific American's Mathematical Games column.^[7] 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.^[14]

Operation

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 e, d, and n, such that with modular exponentiation for all integers m (with 0 ≤ m < n):

and that knowing e and n, it can be extremely difficult to find d. Here the symbol ≡ denotes modular congruence: i.e. both (m^e)^d and m have the same remainder when divided by n.

In addition, for some operations it is convenient that the order of the two exponentiations can be changed: the previous relation also implies

RSA involves a public key and a

. The public key can be known by everyone and is used for encrypting messages. The intention is that messages encrypted with the public key can only be decrypted in a reasonable amount of time by using the private key. The public key is represented by the integers n and e, and the private key by the integer d (although n is also used during the decryption process, so it might be considered to be a part of the private key too). m represents the message (previously prepared with a certain technique explained below).

Key generation

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

1. Choose two large prime numbers p and q.
   • To make factoring harder, p and q should be chosen at random, be both large and have a large difference.^[1] For choosing them the standard method is to choose random integers and use a primality test until two primes are found.
   • p and q should be kept secret.
2. Compute n = pq.
   • n is used as the
     key length
     .
   • n is released as part of the public key.
3. Compute λ(n), where λ is
   φ
   (p) = p − 1, and likewise λ(q) = q − 1. Hence λ(n) = lcm(p − 1, q − 1).
   • The lcm may be calculated through the Euclidean algorithm, since lcm(a, b) = |ab|/gcd(a, b).
   • λ(n) is kept secret.
4. Choose an integer e such that 1 < e < λ(n) and
   coprime
   .
   • e having a short bit-length and small Hamming weight results in more efficient encryption – the most commonly chosen value for e is 2¹⁶ + 1 = 65537. The smallest (and fastest) possible value for e is 3, but such a small value for e has been shown to be less secure in some settings.^[15]
   • e is released as part of the public key.
5. Determine d as d ≡ e⁻¹ (mod λ(n)); that is, d is the modular multiplicative inverse of e modulo λ(n).
   • This means: solve for d the equation de ≡ 1 (mod λ(n)); d can be computed efficiently by using the extended Euclidean algorithm, since, thanks to e and λ(n) being coprime, said equation is a form of Bézout's identity, where d is one of the coefficients.
   • d is kept secret as the private key exponent.

The public key consists of the modulus n and the public (or encryption) exponent e. The private key consists of the private (or decryption) exponent d, which must be kept secret. p, q, and λ(n) must also be kept secret because they can be used to calculate d. In fact, they can all be discarded after d has been computed.^[16]

In the original RSA paper,

(Section B.3.1) may require that d < λ(n). Any "oversized" private exponents not meeting this criterion may always be reduced modulo λ(n) to obtain a smaller equivalent exponent.

Since any common factors of (p − 1) and (q − 1) are present in the factorisation of n − 1 = pq − 1 = (p − 1)(q − 1) + (p − 1) + (q − 1),[17]^{[self-published source?]} it is recommended that (p − 1) and (q − 1) have only very small common factors, if any, besides the necessary 2.^{[1][18][19][failed verification][20][failed verification]}

Note: The authors of the original RSA paper carry out the key generation by choosing d and then computing e as the

Key distribution

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 (n, e) to Bob via a reliable, but not necessarily secret, route. Alice's private key (d) is never distributed.

Encryption

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

To do it, he first turns M (strictly speaking, the un-padded plaintext) into an integer m (strictly speaking, the padded plaintext), such that 0 ≤ m < n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext c, using Alice's public key e, corresponding to

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

Decryption

Alice can recover m from c by using her private key exponent d by computing

Given m, she can recover the original message M by reversing the padding scheme.

Example

Here is an example of RSA encryption and decryption:^[b]

1. Choose two distinct prime numbers, such as

   ${\displaystyle p=61}$ and ${\displaystyle q=53}$.

2. Compute n = pq giving

   ${\displaystyle n=61\times 53=3233.}$

3. Compute the
   Carmichael's totient function of the product as λ(n) = lcm
   (p − 1, q − 1) giving

   ${\displaystyle \lambda (3233)=\operatorname {lcm} (60,52)=780.}$

4. Choose any number 2 < e < 780 that is
   coprime
   to 780. Choosing a prime number for e leaves us only to check that e is not a divisor of 780.

   Let ${\displaystyle e=17}$.

5. Compute d, the modular multiplicative inverse of e (mod λ(n)), yielding
   
   $${\displaystyle d=413,}$$

   
   as ${\displaystyle 1=(17\times 413){\bmod {7}}80.}$

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

The private key is (n = 3233, d = 413). For an encrypted ciphertext c, the decryption function is

For instance, in order to encrypt m = 65, one calculates

To decrypt c = 2790, one calculates

Both of these calculations can be computed efficiently using the

. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor n = 3233 (obtained from the freely available public key) back to the primes p and q. e, also from the public key, is then inverted to get d, 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 d_p, d_q and q_inv, which are part of the private key are computed as follows:

Here is how d_p, d_q and q_inv are used for efficient decryption (encryption is efficient by choice of a suitable d and e pair):

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 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 d (modulo n) (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 e (modulo n) (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:

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 correctness

Proof using Fermat's little theorem

The proof of the correctness of RSA is based on Fermat's little theorem, stating that a^{p − 1} ≡ 1 (mod p) for any integer a and prime p, not dividing a.^{[note 1]}

We want to show that

for every integer m when p and q are distinct prime numbers and e and d are positive integers satisfying ed ≡ 1 (mod λ(pq)).

Since λ(pq) = lcm(p − 1, q − 1) is, by construction, divisible by both p − 1 and q − 1, we can write

for some nonnegative integers h and k.^{[note 2]}

To check whether two numbers, such as m^ed and m, are congruent mod pq, it suffices (and in fact is equivalent) to check that they are congruent mod p and mod q separately.^{[note 3]}

To show m^ed ≡ m (mod p), we consider two cases:

1. If m ≡ 0 (mod p), m is a multiple of p. Thus m^ed is a multiple of p. So m^ed ≡ 0 ≡ m (mod p).
2. If m ${\displaystyle \not \equiv }$ 0 (mod p),

   ${\displaystyle m^{ed}=m^{ed-1}m=m^{h(p-1)}m=(m^{p-1})^{h}m\equiv 1^{h}m\equiv m{\pmod {p}},}$

   where we used Fermat's little theorem to replace m^p−1 mod p with 1.

The verification that m^ed ≡ m (mod q) proceeds in a completely analogous way:

1. If m ≡ 0 (mod q), m^ed is a multiple of q. So m^ed ≡ 0 ≡ m (mod q).
2. If m ${\displaystyle \not \equiv }$ 0 (mod q),

   ${\displaystyle m^{ed}=m^{ed-1}m=m^{k(q-1)}m=(m^{q-1})^{k}m\equiv 1^{k}m\equiv m{\pmod {q}}.}$

This completes the proof that, for any integer m, and integers e, d such that ed ≡ 1 (mod λ(pq)),

Notes

Proof using Euler's theorem

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 m^ed ≡ m (mod n), where n = pq is a product of two different prime numbers, and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)). Since e and d are positive, we can write ed = 1 + hφ(n) for some non-negative integer h. Assuming that m is relatively prime to n, we have

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

More generally, for any e and d satisfying ed ≡ 1 (mod λ(n)), the same conclusion follows from Carmichael's generalization of Euler's theorem, which states that m^λ(n) ≡ 1 (mod n) for all m relatively prime to n.

When m is not relatively prime to n, the argument just given is invalid. This is highly improbable (only a proportion of 1/p + 1/q − 1/(pq) numbers have this property), but even in this case, the desired congruence is still true. Either m ≡ 0 (mod p) or m ≡ 0 (mod q), and these cases can be treated using the previous proof.

Padding

Attacks against plain RSA

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

• When encrypting with low encryption exponents (e.g., e = 3) and small values of the m (i.e., m < n^1/e), the result of m^e is strictly less than the modulus n. In this case, ciphertexts can be decrypted easily by taking the eth root of the ciphertext over the integers.
• If the same clear-text message is sent to e or more recipients in an encrypted way, and the receivers share the same exponent e, but different p, q, and therefore n, then it is easy to decrypt the original clear-text message via the Chinese remainder theorem. Johan Håstad noticed that this attack is possible even if the clear texts are not equal, but the attacker knows a linear relation between them.^[22] This attack was later improved by Don Coppersmith (see Coppersmith's attack).^[23]
• Because RSA encryption is a
  semantically secure if an attacker cannot distinguish two encryptions from each other, even if the attacker knows (or has chosen) the corresponding plaintexts. RSA without padding is not semantically secure.^[24]
• RSA has the property that the product of two ciphertexts is equal to the encryption of the product of the respective plaintexts. That is, m₁^em₂^e ≡ (m₁m₂)^e (mod n). Because of this multiplicative property, a chosen-ciphertext attack is possible. E.g., an attacker who wants to know the decryption of a ciphertext c ≡ m^e (mod n) may ask the holder of the private key d to decrypt an unsuspicious-looking ciphertext c′ ≡ cr^e (mod n) for some value r chosen by the attacker. Because of the multiplicative property, c' is the encryption of mr (mod n). Hence, if the attacker is successful with the attack, they will learn mr (mod n), from which they can derive the message m by multiplying mr with the modular inverse of r modulo n.^{[citation needed]}
• Given the private exponent d, one can efficiently factor the modulus n = pq. And given factorization of the modulus n = pq, one can obtain any private key (d', n) generated against a public key (e', n).^[15]

Padding schemes

To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. 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

).

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 (U.S. patent 6,266,771 and U.S. patent 7,036,014); however, these patents expired on 24 July 2009 and 25 April 2010 respectively. Use of PSS no longer seems to be encumbered by patents.^{[original research?]} Note that using different RSA key pairs for encryption and signing is potentially more secure.^[26]

Security and practical considerations

Using the Chinese remainder algorithm

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.^{[citation needed]} The following values are precomputed and stored as part of the private key:

• ${\displaystyle p}$ and ${\displaystyle q}$ – the primes from the key generation,
• ${\displaystyle d_{P}=d{\pmod {p-1}},}$
• ${\displaystyle d_{Q}=d{\pmod {q-1}},}$
• ${\displaystyle q_{\text{inv}}=q^{-1}{\pmod {p}}.}$

These values allow the recipient to compute the exponentiation m = c^d (mod pq) more efficiently as follows:
     ${\displaystyle m_{1}=c^{d_{P}}{\pmod {p}}}$,
     ${\displaystyle m_{2}=c^{d_{Q}}{\pmod {q}}}$,
     ${\displaystyle h=q_{\text{inv}}(m_{1}-m_{2}){\pmod {p}}}$,^[c]
     ${\displaystyle m=m_{2}+hq}$.

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 problem

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.^[27]

The RSA problem is defined as the task of taking eth roots modulo a composite n: recovering a value m such that c ≡ m^e (mod n), where (n, e) is an RSA public key, and c is an RSA ciphertext. 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 lcm(p − 1, q − 1) that 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.

Multiple polynomial quadratic sieve (MPQS) can be used to factor the public modulus n.

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.

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^[1] that Miller has shown that – assuming the truth of the extended Riemann hypothesis – finding d from n and e is as hard as factoring n into p and q (up to a polynomial time difference).^[29] 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.

As of 2020^[update], the largest publicly known factored

It is generally presumed that RSA is secure if n is sufficiently large, outside of quantum computing.

If n is 300 

In 1994,

.

Faulty key generation

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Finding the large primes p and q 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 p and q should not be "too close", lest the

, and hence such values of p or q should be discarded.

It is important that the private exponent d be large enough. Michael J. Wiener showed that if p is between q and 2q (which is quite typical) and d < n^1/4/3, then d can be computed efficiently from n and e.[34]

There is no known attack against small public exponents such as e = 3, provided that the proper padding is used.

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.

In October 2017, a team of researchers from

Importance of strong random number generation

A cryptographically strong

]

They exploited a weakness unique to cryptosystems based on integer factorization. If n = pq is one public key, and n′ = p′q′ is another, then if by chance p = p′ (but q is not equal to q'), then a simple computation of gcd(n, n′) = p factors both n and n', 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 q given p, instead of choosing p and q 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 n against the product of all the other keys n' they had found (a 729-million-digit number), instead of computing each gcd(n, n′) 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.^[38]

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 attacks

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 c^d (mod n), Alice first chooses a secret random value r and computes (r^ec)^d (mod n). The result of this computation, after applying Euler's theorem, is rc^d (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,

, 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.^[40][41] It impacted the Mozilla NSS Crypto Library, which was used notably by Firefox and Chrome.

Side-channel analysis attacks

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",^[42] 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.^{[43][self-published source?]} The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.

Tricky implementation

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 the book Practical Cryptography With Go suggests avoiding RSA if possible.^[44]

Implementations

Some cryptography libraries that provide support for RSA include:

• Botan
• Bouncy Castle
• cryptlib
• Crypto++
• Libgcrypt
• Nettle
• OpenSSL
• wolfCrypt
• GnuTLS
• mbed TLS
• LibreSSL

See also

• Acoustic cryptanalysis
• Computational complexity theory
• Diffie–Hellman key exchange
• Digital Signature Algorithm
• Elliptic-curve cryptography
• Key exchange
• Key management
• Key size
• Public-key cryptography
• Rabin cryptosystem
• Trapdoor function

Notes

References

Further reading

• Menezes, Alfred; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). Handbook of Applied Cryptography. CRC Press.
  ISBN 978-0-8493-8523-0
  .
• ISBN 978-0-262-03293-3
  .

External links

• 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, U.S. patent 4,405,829.
• PKCS #1: RSA Cryptography Standard (
  RSA Laboratories
  website)
  • The
    primitives; encryption schemes; signature schemes with appendix; ASN.1
    syntax for representing keys and for identifying the schemes".
• Explanation of RSA using colored lamps on
  YouTube
• Thorough walk through of RSA
• Prime Number Hide-And-Seek: How the RSA Cipher Works
• Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: On the Power of Simple Branch Prediction Analysis
• Example of an RSA implementation with PKCS#1 padding (GPL source code) Archived 2012-07-22 at the Wayback Machine
• Kocher's article about timing attacks
• An animated explanation of RSA with its mathematical background by CrypTool
• Grime, James. "RSA Encryption". Numberphile. Brady Haran. Archived from the original on 2018-10-06. Retrieved 2013-04-13.
• How RSA Key used for Encryption in real world