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Suppose Pao-Chi sends an e-mail to Gwen using a digital envelope, and Satomi indeed intercepts the message. Will Satomi be able to read it The
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bulk data was encrypted using a symmetric algorithm, so she needs the session key. To decrypt the data she could try a brute force attack, but if the key is 128 bits, that would take billions or even trillions of millennia (as you saw in 2). But because the session key is right there, part of the message itself, it seems she doesn t need to try this attack except the session key is also encrypted. To decrypt the session key, she needs the partner to the public key that was used to encrypt it because that s the only key that will decrypt it. That s the private key, but only Gwen has that. Maybe Satomi can break the public-key algorithm or perform a brute force attack to find the private key. Recall that there were two ways to recover messages encrypted using a symmetric-key crypto: break the algorithm or find the key using brute force. The same is true for public-key crypto. If Satomi can figure out what the private key is by breaking the algorithm or using brute force, she can decrypt the session key and use it to decrypt the bulk data. To determine the private key, Satomi must finds a 160-bit to 510-bit (or possibly higher) number. If a brute force attack on a 128-bit value (the symmetric key) is outside the realm of feasibility, then so is such an attack on a 160-bit number. So a brute force attack on the 160-bit or 510-bit number is not a realistic option. What about the algorithm Can a public-key algorithm be broken It turns out that all public-key algorithms can be broken by determining what the private key is, based on the public key. Remember that the public and private keys are partners, that they re related, and that this relationship is mathematical. Math computations can be used to derive the private key from the public key. Luckily, these math computations are time-consuming. As with symmetric-key crypto, the longer the public key, the longer it will take to derive the private key from it. If the keys are long enough, solving the math problem would take as much time as a brute force attack on a 96-bit to 128-bit key. In the section titled Key Sizes, we talk about key sizes for public-key algorithms.
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In 2, we say that you should use only symmetric algorithms with no weaknesses that the fastest way to break them should be a brute force attack. Why, then, are we now telling you to use public-key algorithms
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The Key Distribution Problem and Public-Key Cryptography
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that can be broken For these algorithms, the brute force attack is not the fastest attack. Why the change of heart The answer is simple: No one has been able to develop a public-key algorithm that has no weaknesses. For all public-key algorithms, there are techniques that will break them faster than brute force. Think of these techniques as shortcuts. But most users are willing to live with the shortcuts for two reasons. First, cryptographers have performed a tremendous amount of research quantifying the time required by the shortcuts. Even though an algorithm is susceptible to an attack faster than brute force, the research shows it still takes a long time. For most people, that amount of time is sufficient security. Second, people are willing to use algorithms that suffer from shortcuts because these algorithms are the best way to solve the key distribution problem. For people who don t trust public-key cryptography, the only recourse is to use a shared secret scheme for key distribution. Otherwise, until someone comes up with a public-key algorithm with no shortcuts, we ll have to live with them. Actually, though, having the shortcuts is not too bad. Using brute force, an attacker might get lucky and find the key in one of the first few tries, theoretically reducing the time of a successful attack to almost zero. In contrast, cryptographers know how long they can expect it will take to break a public-key algorithm using a shortcut. These attacks usually must run their entire course before coming up with the answer, almost never hitting on a lucky early answer, so researchers have established a more concrete minimum attack time.
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