Author: Minh Van Nguyen <nguyenminh2@gmail.com>
This tutorial uses Sage to study elementary number theory and the RSApublic key cryptosystem. A number of Sage commands will be presentedthat help us to perform basic number theoretic operations such asgreatest common divisor and Eulerâs phi function. We then present theRSA cryptosystem and use Sageâs built-in commands to encrypt anddecrypt data via the RSA algorithm. Note that this tutorial on RSA isfor pedagogy purposes only. For further details on cryptography orthe security of various cryptosystems, consult specialized texts suchas[MenezesEtAl1996],[Stinson2006], and[TrappeWashington2006].
Elementary number theory¶
The Rivest, Shamir, Adleman (RSA) cryptosystem is an example of a public key cryptosystem. RSA uses a public key to encrypt messages and decryption is performed using a corresponding private key. We can distribute our public keys, but for security reasons we should keep our private keys to ourselves.
We first review basic concepts from elementary number theory,including the notion of primes, greatest common divisors, congruencesand Eulerâs phi function. The number theoretic concepts and Sagecommands introduced will be referred to in later sections when wepresent the RSA algorithm.
Prime numbers¶
Public key cryptography uses many fundamental concepts from numbertheory, such as prime numbers and greatest common divisors. Apositive integer (n > 1) is said to be prime if its factors areexclusively 1 and itself. In Sage, we can obtain the first 20 primenumbers using the command
primes_first_n :
Greatest common divisors¶
Let (a) and (b) be integers, not both zero. Then the greatest commondivisor (GCD) of (a) and (b) is the largest positive integer which isa factor of both (a) and (b). We use (gcd(a,b)) to denote thislargest positive factor. One can extend this definition by setting(gcd(0,0) = 0). Sage uses
gcd(a,b) to denote the GCD of (a)and (b). The GCD of any two distinct primes is 1, and the GCD of 18and 27 is 9.
If (gcd(a,b) = 1), we say that (a) is coprime (or relativelyprime) to (b). In particular, (gcd(3, 59) = 1) so 3 is coprime to 59and vice versa.
Congruences¶
When one integer is divided by a non-zero integer, we usually get aremainder. For example, upon dividing 23 by 5, we get a remainder of3; when 8 is divided by 5, the remainder is again 3. The notion ofcongruence helps us to describe the situation in which two integershave the same remainder upon division by a non-zero integer. Let(a,b,n in ZZ) such that (n neq 0). If (a) and (b) have thesame remainder upon division by (n), then we say that (a) iscongruent to (b) modulo (n) and denote this relationship by
This definition is equivalent to saying that (n) divides thedifference of (a) and (b), i.e. (n ;|; (a - b)). Thus(23 equiv 8 pmod{5}) because when both 23 and 8 are divided by 5, weend up with a remainder of 3. The command
mod allows us tocompute such a remainder:
Eulerâs phi function¶
Consider all the integers from 1 to 20, inclusive. List all thoseintegers that are coprime to 20. In other words, we want to findthose integers (n), where (1 leq n leq 20), such that(gcd(n,20) = 1). The latter task can be easily accomplished with alittle bit of Sage programming:
The above programming statements can be saved to a text file called,say,
/home/mvngu/totient.sage , organizing it as follows to enhancereadability.
We refer to
totient.sage as a Sage script, just as one would referto a file containing Python code as a Python script. We use 4 spaceindentations, which is a coding convention in Sage as well as Pythonprogramming, instead of tabs.
The command
load can be used to read the file containing ourprogramming statements into Sage and, upon loading the content of thefile, have Sage execute those statements:
From the latter list, there are 8 integers in the closed interval([1, 20]) that are coprime to 20. Without explicitly generating thelist
how can we compute the number of integers in ([1, 20]) that arecoprime to 20? This is where Eulerâs phi function comes in handy.Let (n in ZZ) be positive. Then Eulerâs phi function counts thenumber of integers (a), with (1 leq a leq n), such that(gcd(a,n) = 1). This number is denoted by (varphi(n)). Eulerâs phifunction is sometimes referred to as Eulerâs totient function, hencethe name
totient.sage for the above Sage script. The commandeuler_phi implements Eulerâs phi function. To compute(varphi(20)) without explicitly generating the above list, we proceedas follows:
How to keep a secret?¶
Cryptography is the science (some might say art) of concealingdata. Imagine that we are composing a confidential email tosomeone. Having written the email, we can send it in one of two ways.The first, and usually convenient, way is to simply press the sendbutton and not care about how our email will be delivered. Sending anemail in this manner is similar to writing our confidential message ona postcard and post it without enclosing our postcard inside anenvelope. Anyone who can access our postcard can see our message.On the other hand, before sending our email, we can scramble theconfidential message and then press the send button. Scrambling ourmessage is similar to enclosing our postcard inside an envelope.While not 100% secure, at least we know that anyone wanting to readour postcard has to open the envelope.
In cryptography parlance, our message is called plaintext. Theprocess of scrambling our message is referred to as encryption.After encrypting our message, the scrambled version is calledciphertext. From the ciphertext, we can recover our originalunscrambled message via decryption. The following figureillustrates the processes of encryption and decryption. Acryptosystem is comprised of a pair of related encryption anddecryption processes.
The following table provides a very simple method of scrambling amessage written in English and using only upper case letters,excluding punctuation characters.
Formally, let
[Sigma={ mathtt{A}, mathtt{B}, mathtt{C}, dots, mathtt{Z} }]
be the set of capital letters of the English alphabet. Furthermore,let
be the American Standard Code for Information Interchange (ASCII)encodings of the upper case English letters. Then the above tableexplicitly describes the mapping (f: Sigma longrightarrow Phi).(For those familiar with ASCII, (f) is actually a common process forencoding elements of (Sigma), rather than a cryptographicâscramblingâ process per se.) To scramble a message written usingthe alphabet (Sigma), we simply replace each capital letter of themessage with its corresponding ASCII encoding. However, thescrambling process described in the above table provides,cryptographically speaking, very little to no security at all and westrongly discourage its use in practice.
Keeping a secret with two keys¶
The Rivest, Shamir, Adleman (RSA) cryptosystem is an example of apublic key cryptosystem. RSA uses a public key toencrypt messages and decryption is performed using a correspondingprivate key. We can distribute our public keys, but forsecurity reasons we should keep our private keys to ourselves. Theencryption and decryption processes draw upon techniques fromelementary number theory. The algorithm below is adapted from page165 of [TrappeWashington2006]. It outlines the RSA procedure forencryption and decryption.
The next two sections will step through the RSA algorithm, usingSage to generate public and private keys, and perform encryptionand decryption based on those keys.
Generating public and private keys¶
Positive integers of the form (M_m = 2^m - 1) are calledMersenne numbers. If (p) is prime and (M_p = 2^p - 1) is alsoprime, then (M_p) is called a Mersenne prime. For example, 31is prime and (M_{31} = 2^{31} - 1) is a Mersenne prime, as can beverified using the command
is_prime(p) . This command returnsTrue if its argument p is precisely a prime number;otherwise it returns False . By definition, a prime must be apositive integer, hence is_prime(-2) returns False although we know that 2 is prime. Indeed, the number(M_{61} = 2^{61} - 1) is also a Mersenne prime. We can use(M_{31}) and (M_{61}) to work through step 1 in the RSA algorithm:
A word of warning is in order here. In the above code example, thechoice of (p) and (q) as Mersenne primes, and with so many digits farapart from each other, is a very bad choice in terms of cryptographicsecurity. However, we shall use the above chosen numeric values for(p) and (q) for the remainder of this tutorial, always bearing in mindthat they have been chosen for pedagogy purposes only. Refer to[MenezesEtAl1996],[Stinson2006], and[TrappeWashington2006]for in-depth discussions on the security of RSA, or consult otherspecialized texts.
For step 2, we need to find a positive integer that is coprime to(varphi(n)). The set of integers is implemented within the Sagemodule
sage.rings.integer_ring . Various operations onintegers can be accessed via the ZZ.* family of functions.For instance, the command ZZ.random_element(n) returns apseudo-random integer uniformly distributed within the closed interval([0, n-1]).
We can compute the value (varphi(n)) by calling the sage function
euler_phi(n) , but for arbitrarily large prime numbers (p) and (q),this can take an enormous amount of time. Indeed, the private keycan be quickly deduced from the public key once you know (varphi(n)),so it is an important part of the security of the RSA cryptosystem that(varphi(n)) cannot be computed in a short time, if only (n) is known.On the other hand, if the private key is available, we can compute(varphi(n)=(p-1)(q-1)) in a very short time.
Using a simple programming loop, we can compute therequired value of (e) as follows:
As
e is a pseudo-random integer, its numeric value changesafter each execution of e=ZZ.random_element(phi) .
To calculate a value for
d in step 3 of the RSA algorithm, we usethe extended Euclidean algorithm. By definition of congruence,(de equiv 1 pmod{varphi(n)}) is equivalent to
[de - k cdot varphi(n) = 1]
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where (k in ZZ). From steps 1 and 2, we already know the numericvalues of (e) and (varphi(n)). The extended Euclidean algorithmallows us to compute (d) and (-k). In Sage, this can be accomplishedvia the command
xgcd . Given two integers (x) and (y),xgcd(x,y) returns a 3-tuple (g,s,t) that satisfiesthe Bézout identity (g = gcd(x,y) = sx + ty). Having computed avalue for d , we then use the commandmod(d*e,phi) to check that d*e is indeed congruentto 1 modulo phi .
Thus, our RSA public key is
[(n, e)=(4951760154835678088235319297, 1850567623300615966303954877)]
and our corresponding private key is
[(p, q, d)=(2147483647, 2305843009213693951, 4460824882019967172592779313)]
Encryption and decryption¶
Suppose we want to scramble the message
HELLOWORLD using RSAencryption. From the above ASCII table, our message maps to integersof the ASCII encodings as given below.
Concatenating all the integers in the last table, our message can berepresented by the integer
[m = 72697676798779827668]
There are other more cryptographically secure means for representingour message as an integer. The above process is used fordemonstration purposes only and we strongly discourage its use inpractice. In Sage, we can obtain an integer representation of ourmessage as follows:
To encrypt our message, we raise (m) to the power of (e) and reducethe result modulo (n). The command
mod(a^b,n) first computesa^b and then reduces the result modulo n . If the exponentb is a âlargeâ integer, say with more than 20 digits, thenperforming modular exponentiation in this naive manner takes quitesome time. Brute force (or naive) modular exponentiation isinefficient and, when performed using a computer, can quicklyconsume a huge quantity of the computerâs memory or result in overflowmessages. For instance, if we perform naive modular exponentiationusing the command mod(m^e,n) , where m , n and e are asgiven above, we would get an error message similar to the following:
There is a trick to efficiently perform modular exponentiation, calledthe method of repeated squaring, cf. page 879 of [CormenEtAl2001].Suppose we want to compute (a^b mod n). First, let(d mathrel{mathop:}= 1) and obtain the binary representation of (b),say ((b_1, b_2, dots, b_k)) where each (b_i in ZZ/2ZZ). For(i mathrel{mathop:}= 1, dots, k), let(d mathrel{mathop:}= d^2 mod n) and if (b_i = 1) then let(d mathrel{mathop:}= da mod n). This algorithm is implemented inthe function
power_mod . We now use the function power_mod toencrypt our message:
Thus (c = 630913632577520058415521090) is the ciphertext. To recoverour plaintext, we raise
c to the power of d and reduce theresult modulo n . Again, we use modular exponentiation viarepeated squaring in the decryption process:
Notice in the last output that the value 72697676798779827668 is thesame as the integer that represents our original message. Hence wehave recovered our plaintext.
Acknowledgements¶
Bibliography¶
RSA (RivestâShamirâAdleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and it is different from the decryption key which is kept secret (private). In RSA, this asymmetry is based on the practical difficulty of the factorization of the product of two large prime numbers, the 'factoring problem'. The acronym RSA is made of the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who first publicly described the algorithm in 1977. Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), had developed an equivalent system in 1973, but this was not declassified until 1997.[1]
A user of RSA creates and then publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers must be kept secret. Anyone can use the public key to encrypt a message, but only someone with knowledge of the prime numbers can decode the message.[2] Breaking RSA encryption is known as the RSA problem. Whether it is as difficult as the factoring problem remains an open question. There are currently no published methods to defeat the system if a large enough key is used.
RSA is a relatively slow algorithm, and because of this, it is less commonly used to directly encrypt user data. More often, RSA passes encrypted shared keys for symmetric key cryptography which in turn can perform bulk encryption-decryption operations at much higher speed.
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.[3]
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 one-way 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.[4] In April 1977, they spent Passover at the house of a student and drank a good deal of Manischewitz wine before returning to their homes at around midnight.[5] 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 â the initials of their surnames in same order as their paper.[6]
Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), described an equivalent system in an internal document in 1973.[7] However, given the relatively expensive computers needed to implement it at the time, RSA was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification.
Kid-RSA (KRSA) is a simplified 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.[8][9][10][11][12]
Patent
MIT was granted for a 'Cryptographic communications system and method' that used the algorithm, on September 20, 1983. Though the patent was going to expire on September 21, 2000 (the term of patent was 17 years at the time), the algorithm was released to the public domain by RSA Security on September 6, 2000, two weeks earlier.[13] Since a detailed description of the algorithm had been published in the Mathematical Games column in the August 1977 issue of Scientific American, 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. Had Cocks's work been publicly known, a patent in the United States would not have been legal either.
From the DWPI's abstract of the patent,
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, and such that with modular exponentiation for all integers (with):
and that even knowing and or even it can be extremely difficult to find . The triple bar (â¡) here denotes modular congruence.
In addition, for some operations it is convenient that the order of the two exponentiations can be changed and that this relation also implies:
(md)eequivmpmod{n}
RSA involves a public key and a private key. The public key can be known by everyone, and it 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 and ; and, the private key, by the integer (although is also used during the decryption process. Thus, it might be considered to be a part of the private key, too). 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:
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.[15]
In the original RSA paper, the Euler totient function is used instead of λ(n) for calculating the private exponent d. Since Ï(n) is always divisible by λ(n) the algorithm works as well. That the Euler totient function can be used can also be seen as a consequence of the Lagrange's theorem applied to the multiplicative group of integers modulo pq. Thus any d satisfying also satisfies . However, computing d modulo Ï(n) will sometimes yield a result that is larger than necessary (i.e.). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent d at all, rather than using the optimized decryption method based on the Chinese remainder theorem described below), but some standards like FIPS 186-4 may require that . Any 'oversized' private exponents not meeting that criterion may always be reduced modulo λ(n) to obtain a smaller equivalent exponent.
Since any common factors of and are present in the factorisation of = =,[16] it is recommended that and have only very small common factors, if any besides the necessary 2.[17][18][19]
Note: The authors of the original RSA paper carry out the key generation by choosing d and then computing e as the modular multiplicative inverse of d modulo Ï(n), whereas most current implementations of RSA, such as those following PKCS#1, do the reverse (choose e and compute d). 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.[20]
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 to Bob via a reliable, but not necessarily secret, route. Alice's private key is never distributed.
Encryption
After Bob obtains Alice's public key, he can send a message to Alice.
To do it, he first turns (strictly speaking, the un-padded plaintext) into an integer (strictly speaking, the padded plaintext), such that by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext, using Alice's public key, corresponding to
This can be done reasonably quickly, even for very large numbers, using modular exponentiation. Bob then transmits to Alice.
Decryption
Alice can recover from by using her private key exponent by computing
cdequiv(me)dequivmpmod{n}
Given, she can recover the original message by reversing the padding scheme.
Example
Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also use OpenSSL to generate and examine a real keypair.
q=53
λ(3233)=operatorname{lcm}(60,52)=780
Let
d=413
Worked example for the modular multiplicative inverse:
413 x 17=1bmod780
The public key is . For a padded plaintext message m, the encryption function is
The private key is . For an encrypted ciphertextc, the decryption function is
m(c)=c413bmod3233
For instance, in order to encrypt, we calculate
To decrypt, we calculate
m=2790413bmod3233=65
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 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 dp, dq and qinv, which are part of the private key are computed as follows:
begin{align}dp={}&dbmod(p-1)=413bmod(61-1)=53dq={}&dbmod(q-1)=413bmod(53-1)=49qinv={}&q-1bmodp=53-1bmod61=38 â {}&(qinv x q)bmodp=38 x 53bmod61=1end{align}
Here is how dp, dq and qinv are used for efficient decryption. (Encryption is efficient by choice of a suitable d and e pair)
begin{align}m1&=
bmodp=279053bmod61=4m2&=
bmodq=279049bmod53=12h&=(qinv x (m1-m2))bmodp=(38 x -8)bmod61=1m&=m2+h x q=12+1 x 53=65end{align}
Code
A working example in JavaScript using BigInteger.js. This code should not be used in production, as
bigInt.randBetween uses Math.random , which is not a cryptographically secure pseudorandom number generator.[21]
'use strict';
/** * RSA hash function reference implementation. * Uses BigInteger.js https://github.com/peterolson/BigInteger.js * Code originally based on https://github.com/kubrickology/Bitcoin-explained/blob/master/RSA.js */const RSA = ;
/** * Generates a k-bit RSA public/private key pair * https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Code * * @param int, bitlength of desired RSA modulus n (should be even) * @returns Result of RSA generation (object with three bigInt members: n, e, d) */RSA.generate = function(keysize) ;
/** * Encrypt * * @param int / bigInt: the 'message' to be encoded * @param int / bigInt: n value returned from RSA.generate aka public key (part I) * @param int / bigInt: e value returned from RSA.generate aka public key (part II) * @returns encrypted message */RSA.encrypt = function(m, n, e) ;
/** * Decrypt * * @param int / bigInt: the 'message' to be decoded (encoded with RSA.encrypt) * @param int / bigInt: d value returned from RSA.generate aka private key * @param int / bigInt: n value returned from RSA.generate aka public key (part I) * @returns decrypted message */RSA.decrypt = function(c, d, n) ;
Signing messagesE&n Railway
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. 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 actual 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.
This works because multiplication is commutative so
(he)d=hed=hde=(hd)eequivhpmod{n}
Thus, the keys may be swapped without loss of generality, that is a private key of a key pair may be used either to:
PaddingAttacks against plain RSA
There are a number of attacks against plain RSA as described below.
See also: Coppersmith's Attack.
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 PKCS#1 have been carefully designed to securely pad messages prior to RSA encryption. 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 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. 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 US patents on PSS were granted (USPTO 6266771 and USPTO 70360140); however, these patents expired on 24 July 2009 and 25 April 2010, respectively. Use of PSS no longer seems to be encumbered by patents. Note that using different RSA key-pairs for encryption and signing is potentially more secure.[25]
Security and practical considerationsUsing the Chinese remainder algorithm
For efficiency many popular crypto libraries (like OpenSSL, Java and .NET) use the following optimization for decryption and signing based on the Chinese remainder theorem. The following values are precomputed and stored as part of the private key:
and
q
: the primes from the key generation, ,
dQ=dpmod{q-1}
and .
These values allow the recipient to compute the exponentiation more efficiently as follows:
m1=
pmod{p}
h=qinv(m1-m2)pmod{p}
(if then some libraries compute h as
qinvleft[left(m1+leftlceil
rightrceilpright)-m2right]pmod{p}
)
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 RSA problem
See also: RSA Factoring Challenge, Integer factorization records and Shor's algorithm. 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.[26]
The RSA problem is defined as the task of taking eth roots modulo a composite n: recovering a value m such that, where 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, then decrypt c using the standard procedure. To accomplish this, an attacker factors n into p and q, and computes 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.
Multiple polynomial quadratic sieve (MPQS) can be used to factor the public modulus n. The time taken to factor 128-bit and 256-bit n on a desktop computer are respectively 2 seconds and 35 minutes.
A tool called YAFU can be used to optimize this process.[27] It took about 5720s to factor 320bit-N on the same computer.
In 2009, Benjamin Moody factored an RSA-512 bit 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 five gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process. The first RSA-512 factorization in 1999 required the equivalent of 8,400 MIPS years, over an elapsed time of about seven months.[28]
Generate Valid Keys E N For The Rsa Cryptosystem Free
Rivest, Shamir, and Adleman noted that Miller has shown that â assuming the truth of the Extended Riemann Hypothesis â 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 equally as hard as factoring.
, the largest factored RSA number was 795 bits long (240 decimal digits, see RSA-240). Its factorization, by a state-of-the-art distributed implementation, took around 900 CPU years. No larger RSA key is known publicly to have been factored. In practice, RSA keys are typically 1024 to 4096 bits long. Some experts believe that 1024-bit keys may become breakable in the near future or may already be breakable by a sufficiently well-funded attacker, though this is disputable. Few people 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 300 bits or shorter, it can be factored in a few hours in 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.[30] A theoretical hardware device named TWIRL, described by Shamir and Tromer in 2003, called into question the security of 1024 bit keys. It is currently recommended that n be at least 2048 bits long.[31]
In 1994, Peter Shor showed that a quantum computer â if one could ever be practically created for the purpose â would be able to factor in polynomial time, breaking RSA; see Shor's algorithm.
Faulty key generation![]()
See also: Coppersmith's Attack and Wiener's Attack.
Finding the large primes p and q is usually done by testing random numbers of the right 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 Fermat factorization for n be successful. If p â q is less than 2n1/4 (n = p * q, which for even small 1024-bit values of n is) solving for p and q is trivial. Furthermore, if either p â 1 or q â 1 has only small prime factors, n can be factored quickly by Pollard's p â 1 algorithm, and such values of p or q should hence 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, then d can be computed efficiently from n and e.[32]
There is no known attack against small public exponents such as, provided that the proper padding is used. Coppersmith's Attack has many applications in attacking RSA specifically if the public exponent e is small and if the encrypted message is short and not padded. 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.
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. Large number of smart cards and TPMs were shown to be affected. Vulnerable RSA keys are easily identified using a test program the team released.[33]
Importance of strong random number generation
A cryptographically strong random number generator, which has been properly seeded with adequate entropy, must be used to generate the primes p and q. 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.[34][35]
They exploited a weakness unique to cryptosystems based on integer factorization. If is one public key and is another, then if by chance (but q is not equal to qâ²), then a simple computation of factors both n and nâ², 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 nâ² they had found (a 729 million digit number), instead of computing each gcd(n,nâ²) 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 over 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 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.[36]
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
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 d 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)[37] 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, Alice first chooses a secret random value r and computes . The result of this computation after applying Euler's Theorem is 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). 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. 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.
Side-channel analysis attacksGenerate Valid Keys E N For The Rsa Cryptosystem Lyrics
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',[38] 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 has been described in 2010.[39] The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.
Rainbow tables attacks
The generated primes can be attacked by rainbow tables because the random numbers are fixed and finite sets.[40]
Generate Valid Keys E N For The Rsa Cryptosystem 2017Implementations
Below are some cryptography libraries that provide support for RSA:
Generate Valid Keys E N For The Rsa Cryptosystem ProgramSee also
Further reading
External links
Notes and References
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article 'RSA (cryptosystem)'.
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