31. Set up an ElGamal signature scheme for Alice.

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a. Assume that Alice pre-processes her messages with a hash function before signing them. Generate an ElGamal signed message from Alice. Playing the role of Bob, verify the signed message.

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PUBLIC K E Y S Y S T E M S

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b. Now, assume that Alice does not prc-process her messages before signing them. Playing the role of the attacker Trudy, forge n signed message.

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32. If Trudy can guess the session key k in the ElGamal signature scheme, she can recover Alice s secret key a. Construct a specific example that illustrates this implcmentation attack, as discussed in Section 6.8.1. 33. In the ElGarrial signature scheme, a different session key must be used for each signed message. If k is used to sign multiple messages, then an implementation attack can be initiated, as described in Section 6.8.1. Construct an example to illustrate this attack.

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7

Public Key Attacks

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There is always more spirit in attack than in defence. - Titus Livius

Introduction

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In this chapter, we cover some attacks on public key systems in detail. The most widely used public key cryptosystems rely on the difficulty of factoring (RSA) and the discrete log problem (Diffie-Hellman and ElGamal). So we first discuss factoring algorithms and algorithms for solving the discrete log problem. These represent fundamental attacks on the underpinnings of the most widely used public key systems. These attacks are roughly the public key equivalents of an exhaustive key search attack on a symmetric cipher. Then we present a fascinating set of attacks on RSA that are, in a sense, the polar opposite of factoring, since these do not directly attack the RSA algorithm. Instead, these attacks take advantage of implementation issues that, under some circumstances, allow an attacker to recover the private key without breaking the RSA algorithm per se. First, we discuss three different timing attacks on RSA. These attacks are examples of side-channel attacks, where an attacker gains information about an underlying computation which, in turn, leaks information about the key. Such attacks have dramatically changed the nature of crypt,analysis and the development of cryptography in

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We also discuss a devastating glitching attack on RSA, where a single induced error can enable an attacker to recover the private key. This attack, which is an example of a fault induction attack [107], is amazing and amazingly simple. Implementation attacks have more than proven their value in the cryptanalysis of smartcards. In fact, in any scenario where the attacker has physical

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PUBLIC K E Y ATTACKS

a w e s to the crypto device holding the key, such attacks are a serious threat. Consequently, these attacks are certain to play a role in the emerging field of trusted computing [4, 481. A practical timing attack has recently been developed which can be used to recover an RSA private key from a Web server [22]. This particular attack is covered in Section 7.4.

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Factoring Algorithms

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The o bvjo u$ math ern a tical breakthrough would be cleveloprricnt of ari eas,y n a y to factor large prime iiumbers. - Bill Gates [56]

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The RSA public key cryptosystem is the gold standard by which all other public key systems are measured. The security of RSA is thought to rest squarely on the difficulty of factoring large integers. More precisely, given N , where N = py, with p and q prime, if we can determine p or q , then we can break RSA. Consequently, a tremendous amount of effort has been devoted to developing efficient factoring methods. In this section we consider several integer factorization methods. First, we briefly discuss the obvious approach, that is, trial division by numbers up to Then we present Dixon s Algorithm, followed by the quadratic sieve, which is a refinement of Dixon s Algorithni. Like trial division, these algorithnis are guaranteed to find the factors of N , provided enough computing power is available. The quadratic sieve is the best available factoring algorithm for numbers having about 110 decimal digits or less, arid it has been used to successfully factor numbers with about 130 decimal digits. For integers with more than about 110 decimal digits, the number field sieve reigns supreme, and we briefly mention this more complex factoring method a t the crid of this section.

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