Elliptic Curve Cryptography
In this article we provide an introduction to elliptic and hyperelliptic curve cryptosystems, which is one of the hot topic in computer data security. These are two important classes of public key cryptosystems, which can be implemented efficiently over a variety of platforms, starting right from small handheld devices like smart cards and mobile phones to large capacity work-stations. All known public key cryptosystems (PKC) derive their security from some computationally hard mathematical problems. The most popular ones are based mainly over two such problems:
1. Integer factorization problem,
2. Discrete logarithm problem.
The RSA cryptosystem is based on the hardness of integer factorization problem. Elliptic and hyperelliptic curve cryptosystems are based on discrete logarithm problem over their respective groups. Let G be a large cyclic multiplicatively written group with generator g.. Given any element y = gαin the group, computing α is called the discrete logarithm problem (DLP) over G
. Many groups have been proposed in literature over which the DLP is believed to be hard. Once such a group is fixed, an ElGamal type cryptosystem can be built over it.
In 1985, Koblitz and Miller independently proposed the set of elliptic curve points over a finite field to be used for cryptographic purposes. Prior to that, elliptic curves had a wide literature developed for over a century by mathematicians purely to satisfy their aesthetic sense. Only recently various applications of elliptic curves are being used in many applied areas like coding theory, number theoretic algorithms for integer factorization and primality testing and cryptography. The main advantage of elliptic curve cryptography is that for suitably chosen curves there is no known subexponential algorithm (like Number Field Sieve (NFS) algorithm for integer factorization) to solve ECDLP (elliptic curve discrete logarithm problem). This leads to smaller key length in comparison to other PKC’s like RSA and other systems based on DLP with equivalent security. The smaller key length, in turn, leads to faster encryption and decryption, savings in bandwidth and efficient implementation.
Thus ECC is a very suitable cryptosystem for resource constrained devices like smart cards and other mobile and wireless devices.
In 1987, Koblitz showed the Jacobians of hyperelliptic curves over finite fields to be a rich source of finite cyclic groups suitable for cryptographic use. Hyperelliptic curves are elliptic curves of higher genera. As with ECDLP, there is no known subexponential time algorithm to solve HCDLP for suitably chosen curve parameters (specifically for curves of lower genera). So hyperelliptic curve cryptosystems (HECC) have the same advantages as ECC. Moreover, in HECC the underlying finite field is smaller and there are many more curves to chose from. However, Addleman and Huang quickly showed that curves of higher genera are not suitable for cryptographic use.
Also, arithmetic in HECC is much more complex than that in ECC. Due to this inherent complexity in the group operations in the Jacobian of the hyperelliptic curves, HECC, with all its advantages has not yet been able to establish itself as a feasible commercial solution to current cryptographic needs. Several groups of researchers in different parts of the globe are working in this area in order to successfully employ hyperelliptic curves in the implementation of various cryptographic primitives.
Side-Channel Attacks
Initially, cryptanalysis was in the realm of mathematicians. The underlying computationally hard problems, which provide the security to each of the public key cryptographic schemes, pose a challenge to the genius of mathematicians. In 1996, Kocher [80] changed the cryptanalytic scenario. He extended cryptanalysis to application domain from the mathematical domain. In this path breaking work, he proposed side channel attacks (SCA). SCA instead of attacking the underlying hard problem, attacks the specific implementation by measuring and analyzing side channel information like timing, power consumption and electro-magnetic (EM) radiation traces of the computation. Side-channel attacks are believed to be the most potential threat to ECC and HECC, which are employed in hand-held mobile devices and used in hostile outdoor environments.
Thus modern security threat to ECC and HECC are two-fold: one based on the generic cryptanalytic attacks and the others based on side-channel attacks. The generic attacks can be avoided by carefully choosing the security parameters of the system, like the curve itself, the underlying field and its representation, the group order, the cofactor, the base point etc. Any naive implementation which neglects security aspects of these parameters can be broken by the generic attacks. For example, if the group order is large, but is divisible by only small primes, the DLP can be broken by Pohlig-Hellman attack.
The second threat to ECC and HECC is from side-channel attacks. As mentioned earlier, these were first proposed by Kocher in 1996. A stronger power based attack was proposed by Kocher et al . Many attacks based on analysis of side-channel information were proposed by various authors. Kocher, in his original work, had proposed timing attacks, which works on the basis of analysis of the timing of computation of various protocols in the system. Okeya and Sakurai observed that even if a system is secure against timing attacks, it may fail to withstand power attacks. In the authors have observed that systems secure against power analysis attacks may not be secure against EM based attacks. Goubin proposed a new attack based on naive choice of the base point. In the authors have discussed that fault analysis can be used for side-channel attacks and countermeasures.
To immunize a cryptosystem from SCA the various algorithms and protocols constituting the cryptosystem are to be designed carefully. Many proposals have come from the research community to immunize ECC and HECC against SCA. In the design of any cryptosystem, efficiency and security are two main concerns.
ΩΩΩ
Sunday, July 5, 2009
Tuesday, June 30, 2009
The Most Significant Result in Modern Cryptography
Introduction
Cryptography is the art of secret writing. It comes from two greek words: cryptos means secret and graphein, whose meaning is writing. Cryptography is used to communicate secretly. It has a long history. In ancient times, it was the science of statecraft. Only kings and emperors used cryptography to communicate with their generals fighting on the frontiers. History says, the great Roman emperor Julius Caesar had devised and used a classical cryptographic scheme, which is known as Shift Cipher or Caesar Cipher, named after him. We will describe the Shift Cipher later in this article. In this age of the Internet, when most of the people in advanced societies use the Internet for their day-to-day affairs (in banking, shopping, paying bills, booking travel and other services etc ) cryptography has become indispensable for all. All of us use a healthy dose of cryptography (mostly unknowingly) in our everyday lives. There are softwares called sniffers, which can be used to grab every piece of information entering and leaving a computer system connected to the Internet. While using the web-based services, if the personal and financial information of the user is sent in plain English, it is highly likely that they will fall in the hands of unscrupulous persons, which can be very detrimental to their financial status. The information superhighway (the Internet) is free-for-all. The information sent over an open channel like the Internet must be somehow garbled so that even if it falls in the hands of some unintended recipients, the real meaning of the message remains undisclosed. Confidentiality is a must in e-banking, e-commerce and services like that. Also, the intended recipient must be able to correctly decipher the meaning of the message. Cryptography is the art (or science) of garbling and ungarbling of messages.
In cryptographic literature, an ungarbled message is called plaintext. The process of garbling is called encryption and the garbled message is called the ciphertext. The process of getting the plaintext from the cipphertext (opposite of encryption) is called decryption. The process of encryption is called a cryptographic scheme. A cryptographic scheme is said to be broken or compromised, if an unintended person can decrypt a ciphertext with practical amount of efforts.
Cryptography uses keys. A cryptographic key in most cases is a number (not a metallic object like the keys used to open doors). Of course this is not a novelty. The number locks also use a secret combination of numbers to unlock them. In cryptography, the numeric keys are used to encrypt and decrypt messages. In Figure 1, we have explained how a sender encrypts the plaintext and the receiver decrypts it. The keys are secret. If the keys are disclosed or if an adversary can decrypt ciphertext messages even without the knowledge of the required cryptographic key then the cryptographic scheme is said to be broken. Breaking of cryptographic schemes is another challenging (and hence interesting) area of study, known as Cryptanalysis.
The classical cryptography is called secret key cryptography or symmetric key cryptography. That is because the cryptographic keys used for encryption and decryption is the same (so symmetric key). The Shift Cipher used by Julius Caesar is a symmetric key encryption scheme.
Classical Cryptography
In this section we will describe the simple Shift Cipher as an example of symmetric key cipher. In English alphabet there are 26 letters. Let us write these letters on the circumference of a circle. We will call this circle the letter-wheel. Before sending a message the sender agrees upon a common key with the receiver. Suppose the key is 5 and the sender wishes to send the message “ROMEO LOVES JULIET” to the receiver. The sender uses the letter-wheel. He takes one letter of the message at a time and starting from the letter's position on the letter-wheel moves five positions in the clockwise direction and picks up that letter. He replaces the original letter in the message by the picked-up letter. The sender repeats this procedure for all the letters in the message. For example R goes to W, O goes to T, M goes to R, E goes to J and O goes to T. Thus the word ROMEO in plaintext becomes WTRJT in the ciphertext. LOVES becomes QTAJX and JULIET becomes OZQNJS. The encrypted message is thus:
WTRJT QTAJX OZQNJS.
The ciphertext looks really funny and does not offer any clue to an unintended receiver about its real meaning. However, the receiver can easily decipher it. He knows the secret key, i.e. 5. He also uses the letter-wheel. He shifts every letter in the ciphertext 5 units in the counterclockwise direction in the letter-wheel, so that W goes to R, T goes O, R goes to M and so on. Thus WTRJT decrypts to ROMEO and thus the whole message can be deciphered. Even if someone intercepts the message, it is difficult for one to decipher it without the knowledge of the secret key, i.e. 5.
There are many such schemes in classical cryptography. Also, some of them are broken now-a-days. Shift cipher in the modern times is not at all secure. Observe that, there are only 26 possible keys. One can try each one of them to find out the exact meaning of the cipher text. Such attacks are called Exhaustive Search Attacks in cryptographic literature. If the key space is small, exhaustive search can be very effective in breaking an encryption scheme. If one uses a computer, the shift cipher can be broken in the fraction of a second.
This does not mean that all symmetric key ciphers are insecure. In the great wars, classical cryptographic schemes played very important roles. All the fighting powers had devised their own encryption schemes. Cryptanalysts of the warring nations burnt much midnight oil while trying to break the opponents' encryption schemes. Many interesting stories are there about the brain games played by the cryptographers of the time. Modern cryptography uses many symmetric key schemes, which are designed using many sophisticated techniques of computer science and mathematics. They are not only secure, but also very fast in encryption and decryption functions. Many schemes can be implemented in hardware making them even faster. In fact many commercial cryptographic devices now use symmetric key cryptography. For example your phone and fax machines may be encrypting your messages before sending them over the cables. If the encryption and decryption processes used in phones are not very efficient then a smooth communication can not take place.
However, description of modern symmetric key schemes is outside the scope of this article. Our intention is to describe, what in our opinion, is the most significant discovery in modern cryptography.
One main disadvantage of symmetric key schemes is key exchange between the sender and the receiver. They must agree upon a common secret key for encryption and decryption and this must be sent by one party to the other very securely. This could be cumbersome and expensive and in most cases may not worth it. Also the classical cryptography believed in “security in obscurity”. That is confidentiality of the message can be maintained if the encryption scheme and the keys are kept secret. Gradually a new school of thought emerged, which thought otherwise. To them nothing but the keys should be confidential. If the encryption method is available in the public domain for scrutiny, its weaknesses can be explored in depth and insecure schemes can be put to rest. An obscure and insecure encryption scheme can provide a false sense of security, which could be very dangerous.
The Most Significant Discovery
In the late seventies, two scientists put forward a very innovative scheme for key agreement. Two people communicating through a insecure channel like the Internet can securely communicate with each other to share a secret key. Even if their communication messages are intercepted by an adversary, the confidentiality of the key can be maintained. The scheme uses a simple law of powers: for any three numbers a, b and c, (a^b)^c = a^(bc) = (a^c)^b and Assumption 1 described below.
These two people are Whitefield Diffie and Martin Hellman. They published their discovery in 1977, which is popularly known as Diffie-Hellman Key Exchange Method. The aim of the present article is to present a glimpse of the novelty of the scheme without going into detailed mathematical rigor.
All of us know that 2^1 = 2, 2^2 = 4, 2^3 = 8 and so on. All of us know the small powers of 2. Also for a small number, it is easy to answer questions like what power of 2 is 64 or 256. But what power of 2 is 4294967296? The answer is 32, which is not very obvious. Of course if one uses a calculator, one can answer this question in a few minutes. Let us consider the 30 digit number 43517237482183718377817827882196. What power of 2 is 43517237482183718377817827882196? It is difficult to answer this question even with a calculator. What if we consider a 300 digit number? If we are considering powers of 2, then the situation is simpler. What, if we consider 300 digit number power of a 300 digit number?
Of course, a computer’s memory does not allow use of arbitrarily large numbers. So it is not possible to use arbitrarily large powers of arbitrarily large numbers. To trim these numbers into manageable size, cryptographers use so called modular arithmetic. Modular arithmetic can be explained by the use of a number-wheel. Use of modular arithmetic increases the complexity of this problem manifold. As this article is intended to present just a flavor, we don’t go into the mathematical details. In order to avoid the mathematical rigor, let us assume:
Assumption 1: There are large numbers a and x such that even if a^x and a are known, it is extremely difficult to find the number x.
Suppose two people wish to communicate between them confidentially. Cryptographers love to use 2 fictitious characters: Alice and Bob. They are far apart and are connected by an insecure channel like the Internet. They wish to share a common key to encrypt and decrypt messages between them. They can do so as described below.
1. First of all they agree upon a large number a. We don’t mind if anybody intercepts the messages between them and comes to know about a. So they can communicate ‘a’ freely between them in plaintext.
2. Alice then arbitrarily chooses one number p and calculates a^p and sends it to Bob. She keeps p secret.
3. Bob also selects a number q arbitrarily, calculates a^q and sends it to Alice. Bob also does not disclose q.
4. After receiving a^q from Bob, Alice raises this number to a power of her secret number p. The number she gets is (a^q)^p = a^(pq).
5. After receiving (a^p) from Alice, Bob raises this number to the power q, his secret. He gets (a^p)^q = a^(pq).
And see, they both have the same number now, which they can use as their secret key to exchange messages using secret key cryptography!!
If an adversary is intercepting their communications, can he find out the secret key a^(pq)? Certainly not. By intercepting Alice’s messages, he can come to know about a and a^p, but he can not find out p due to Assumption 1. Similarly he can not get q. So it is extremely difficult for him to find a^(pq).
This elegant technique was discovered by Diffie and Hellman in 1977 and is popularly known as Diffie-Hellman key exchange algorithm. This led to the birth of an important branch of cryptography, called public key cryptography (PKC) or asymmetric key cryptography. In symmetric key cryptography, one secret key is used for both encryption and decryption. In PKC every user has 2 keys: a public key and a private key. A user's public key is available in the public domain like one's email address. Anybody can send an encrypted message to the user using the user's public key for encryption. User keeps the private key secret. He/she decrypts the message using his/her private key (like password in the email system, only the person knowing the password can access the message). If the private key is compromised, then the system breaks, no more able to provide confidentiality. One major advantage of PKC over symmetric key cryptography is ensuring non-repudiation. Non-repudiation is: after sending a message one should not be able to deny one's action. In non-electronic communication this is ensured by signature. One can not deny a message that he has signed. In electronic communication, this is ensured using digital signatures. Public key cryptography makes it possible to implement digital signatures.
Public key cryptography has played a major role in the success of the Internet. Diffie-Hellman Key Exchange led to birth of these schemes. That is the reason why we designate this as the most important result in modern cryptography.
Cryptography is the art of secret writing. It comes from two greek words: cryptos means secret and graphein, whose meaning is writing. Cryptography is used to communicate secretly. It has a long history. In ancient times, it was the science of statecraft. Only kings and emperors used cryptography to communicate with their generals fighting on the frontiers. History says, the great Roman emperor Julius Caesar had devised and used a classical cryptographic scheme, which is known as Shift Cipher or Caesar Cipher, named after him. We will describe the Shift Cipher later in this article. In this age of the Internet, when most of the people in advanced societies use the Internet for their day-to-day affairs (in banking, shopping, paying bills, booking travel and other services etc ) cryptography has become indispensable for all. All of us use a healthy dose of cryptography (mostly unknowingly) in our everyday lives. There are softwares called sniffers, which can be used to grab every piece of information entering and leaving a computer system connected to the Internet. While using the web-based services, if the personal and financial information of the user is sent in plain English, it is highly likely that they will fall in the hands of unscrupulous persons, which can be very detrimental to their financial status. The information superhighway (the Internet) is free-for-all. The information sent over an open channel like the Internet must be somehow garbled so that even if it falls in the hands of some unintended recipients, the real meaning of the message remains undisclosed. Confidentiality is a must in e-banking, e-commerce and services like that. Also, the intended recipient must be able to correctly decipher the meaning of the message. Cryptography is the art (or science) of garbling and ungarbling of messages.
In cryptographic literature, an ungarbled message is called plaintext. The process of garbling is called encryption and the garbled message is called the ciphertext. The process of getting the plaintext from the cipphertext (opposite of encryption) is called decryption. The process of encryption is called a cryptographic scheme. A cryptographic scheme is said to be broken or compromised, if an unintended person can decrypt a ciphertext with practical amount of efforts.
Cryptography uses keys. A cryptographic key in most cases is a number (not a metallic object like the keys used to open doors). Of course this is not a novelty. The number locks also use a secret combination of numbers to unlock them. In cryptography, the numeric keys are used to encrypt and decrypt messages. In Figure 1, we have explained how a sender encrypts the plaintext and the receiver decrypts it. The keys are secret. If the keys are disclosed or if an adversary can decrypt ciphertext messages even without the knowledge of the required cryptographic key then the cryptographic scheme is said to be broken. Breaking of cryptographic schemes is another challenging (and hence interesting) area of study, known as Cryptanalysis.
The classical cryptography is called secret key cryptography or symmetric key cryptography. That is because the cryptographic keys used for encryption and decryption is the same (so symmetric key). The Shift Cipher used by Julius Caesar is a symmetric key encryption scheme.
Classical Cryptography
In this section we will describe the simple Shift Cipher as an example of symmetric key cipher. In English alphabet there are 26 letters. Let us write these letters on the circumference of a circle. We will call this circle the letter-wheel. Before sending a message the sender agrees upon a common key with the receiver. Suppose the key is 5 and the sender wishes to send the message “ROMEO LOVES JULIET” to the receiver. The sender uses the letter-wheel. He takes one letter of the message at a time and starting from the letter's position on the letter-wheel moves five positions in the clockwise direction and picks up that letter. He replaces the original letter in the message by the picked-up letter. The sender repeats this procedure for all the letters in the message. For example R goes to W, O goes to T, M goes to R, E goes to J and O goes to T. Thus the word ROMEO in plaintext becomes WTRJT in the ciphertext. LOVES becomes QTAJX and JULIET becomes OZQNJS. The encrypted message is thus:
WTRJT QTAJX OZQNJS.
The ciphertext looks really funny and does not offer any clue to an unintended receiver about its real meaning. However, the receiver can easily decipher it. He knows the secret key, i.e. 5. He also uses the letter-wheel. He shifts every letter in the ciphertext 5 units in the counterclockwise direction in the letter-wheel, so that W goes to R, T goes O, R goes to M and so on. Thus WTRJT decrypts to ROMEO and thus the whole message can be deciphered. Even if someone intercepts the message, it is difficult for one to decipher it without the knowledge of the secret key, i.e. 5.
There are many such schemes in classical cryptography. Also, some of them are broken now-a-days. Shift cipher in the modern times is not at all secure. Observe that, there are only 26 possible keys. One can try each one of them to find out the exact meaning of the cipher text. Such attacks are called Exhaustive Search Attacks in cryptographic literature. If the key space is small, exhaustive search can be very effective in breaking an encryption scheme. If one uses a computer, the shift cipher can be broken in the fraction of a second.
This does not mean that all symmetric key ciphers are insecure. In the great wars, classical cryptographic schemes played very important roles. All the fighting powers had devised their own encryption schemes. Cryptanalysts of the warring nations burnt much midnight oil while trying to break the opponents' encryption schemes. Many interesting stories are there about the brain games played by the cryptographers of the time. Modern cryptography uses many symmetric key schemes, which are designed using many sophisticated techniques of computer science and mathematics. They are not only secure, but also very fast in encryption and decryption functions. Many schemes can be implemented in hardware making them even faster. In fact many commercial cryptographic devices now use symmetric key cryptography. For example your phone and fax machines may be encrypting your messages before sending them over the cables. If the encryption and decryption processes used in phones are not very efficient then a smooth communication can not take place.
However, description of modern symmetric key schemes is outside the scope of this article. Our intention is to describe, what in our opinion, is the most significant discovery in modern cryptography.
One main disadvantage of symmetric key schemes is key exchange between the sender and the receiver. They must agree upon a common secret key for encryption and decryption and this must be sent by one party to the other very securely. This could be cumbersome and expensive and in most cases may not worth it. Also the classical cryptography believed in “security in obscurity”. That is confidentiality of the message can be maintained if the encryption scheme and the keys are kept secret. Gradually a new school of thought emerged, which thought otherwise. To them nothing but the keys should be confidential. If the encryption method is available in the public domain for scrutiny, its weaknesses can be explored in depth and insecure schemes can be put to rest. An obscure and insecure encryption scheme can provide a false sense of security, which could be very dangerous.
The Most Significant Discovery
In the late seventies, two scientists put forward a very innovative scheme for key agreement. Two people communicating through a insecure channel like the Internet can securely communicate with each other to share a secret key. Even if their communication messages are intercepted by an adversary, the confidentiality of the key can be maintained. The scheme uses a simple law of powers: for any three numbers a, b and c, (a^b)^c = a^(bc) = (a^c)^b and Assumption 1 described below.
These two people are Whitefield Diffie and Martin Hellman. They published their discovery in 1977, which is popularly known as Diffie-Hellman Key Exchange Method. The aim of the present article is to present a glimpse of the novelty of the scheme without going into detailed mathematical rigor.
All of us know that 2^1 = 2, 2^2 = 4, 2^3 = 8 and so on. All of us know the small powers of 2. Also for a small number, it is easy to answer questions like what power of 2 is 64 or 256. But what power of 2 is 4294967296? The answer is 32, which is not very obvious. Of course if one uses a calculator, one can answer this question in a few minutes. Let us consider the 30 digit number 43517237482183718377817827882196. What power of 2 is 43517237482183718377817827882196? It is difficult to answer this question even with a calculator. What if we consider a 300 digit number? If we are considering powers of 2, then the situation is simpler. What, if we consider 300 digit number power of a 300 digit number?
Of course, a computer’s memory does not allow use of arbitrarily large numbers. So it is not possible to use arbitrarily large powers of arbitrarily large numbers. To trim these numbers into manageable size, cryptographers use so called modular arithmetic. Modular arithmetic can be explained by the use of a number-wheel. Use of modular arithmetic increases the complexity of this problem manifold. As this article is intended to present just a flavor, we don’t go into the mathematical details. In order to avoid the mathematical rigor, let us assume:
Assumption 1: There are large numbers a and x such that even if a^x and a are known, it is extremely difficult to find the number x.
Suppose two people wish to communicate between them confidentially. Cryptographers love to use 2 fictitious characters: Alice and Bob. They are far apart and are connected by an insecure channel like the Internet. They wish to share a common key to encrypt and decrypt messages between them. They can do so as described below.
1. First of all they agree upon a large number a. We don’t mind if anybody intercepts the messages between them and comes to know about a. So they can communicate ‘a’ freely between them in plaintext.
2. Alice then arbitrarily chooses one number p and calculates a^p and sends it to Bob. She keeps p secret.
3. Bob also selects a number q arbitrarily, calculates a^q and sends it to Alice. Bob also does not disclose q.
4. After receiving a^q from Bob, Alice raises this number to a power of her secret number p. The number she gets is (a^q)^p = a^(pq).
5. After receiving (a^p) from Alice, Bob raises this number to the power q, his secret. He gets (a^p)^q = a^(pq).
And see, they both have the same number now, which they can use as their secret key to exchange messages using secret key cryptography!!
If an adversary is intercepting their communications, can he find out the secret key a^(pq)? Certainly not. By intercepting Alice’s messages, he can come to know about a and a^p, but he can not find out p due to Assumption 1. Similarly he can not get q. So it is extremely difficult for him to find a^(pq).
This elegant technique was discovered by Diffie and Hellman in 1977 and is popularly known as Diffie-Hellman key exchange algorithm. This led to the birth of an important branch of cryptography, called public key cryptography (PKC) or asymmetric key cryptography. In symmetric key cryptography, one secret key is used for both encryption and decryption. In PKC every user has 2 keys: a public key and a private key. A user's public key is available in the public domain like one's email address. Anybody can send an encrypted message to the user using the user's public key for encryption. User keeps the private key secret. He/she decrypts the message using his/her private key (like password in the email system, only the person knowing the password can access the message). If the private key is compromised, then the system breaks, no more able to provide confidentiality. One major advantage of PKC over symmetric key cryptography is ensuring non-repudiation. Non-repudiation is: after sending a message one should not be able to deny one's action. In non-electronic communication this is ensured by signature. One can not deny a message that he has signed. In electronic communication, this is ensured using digital signatures. Public key cryptography makes it possible to implement digital signatures.
Public key cryptography has played a major role in the success of the Internet. Diffie-Hellman Key Exchange led to birth of these schemes. That is the reason why we designate this as the most important result in modern cryptography.
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