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Encrypt message M with Alice s public key: C
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={M}~li~~. = [C]*lice.
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Decrypt ciphertext C with Alice s private key: M
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Hashing is not just. for efficiency-it is actually necessary for the security of many sigriat,ure schemes. For example, the ElGamal signat.ure scheme discussed in Section 6.8 is insecure if the message is not hashed before signing. In this section, we ignore the security implications of hashing before signing.
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5.1 INTRODUCTION
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Signing and decrypting are the same operations, so the notation for Alice signing message M is S = [M]Alice, where S is the signed message.
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Encryption and decryption are inverse operations so that
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[{M)AlicelAlice {[MIAlice)Alice = kf.
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Since Alice s public key is public, anyone can compute C = { M } ~ l i ~ ~ . However, only Alice can compute 111 = [ C ] ~ lor ~ ~ signature S = [M]*lice, i the as Alice is assumed to be in sole possession of her private key. That is, anyone can encrypt a message for Alice, but only Alice can decrypt the ciphertext. Furthermore, only Alice can sign a message but anyone can verify the signature by using Alice s public key. Suppose that M is a message that Alice wants to sign. Then Alice could compute the signature as S = [ M ] ~ l and ~ i ~ send S and 1 1 to Bob. Suppose 1 Bob receives S and M , which may or may not equal S and M , respectively. Then Bob checks whether M = {S }Alice, and, if so, he has verified the integrity of the received message. That is, Bob knows with virtual certainty that M = M . While it is also possible to provide integrity using symmetric key cryptography (see the discussion of MAC in Section 4.2 and the discussion of HMAC, below), digital signatures provide integrity and non-repudiation. With symmetric keys, both Alice and Bob share the key, so Alice can claim that Bob forged the integrity operation, and thereby repudiate the message. But Alice cannot repudiate a message she digitally signed, since only she has access to her private key. However, private key operations are costly to compute and sending both S and M requires twice as much bandwidth as sending M . To reduce the bandwidth usage, Alice could instead compute S = [ h ( M ) ] A l i c e and send M and this small S to Bob. In this case, when Bob receives M and S he must verify that h(M ) = { S t } ~ l i c e . Assuming that hashing is more efficient than private key operations, we not only save bandwidth, but we also increase signing efficiency. In fact, hashing is orders of magnitude more efficient than private key operations. Consequently, this method of signing the hash of M is virtually always used in practice. However, it is important to note that by signing the hash, the security of the signature now depends not only on the security of the public key system, but also on the security of the hash function. To see why this is so, suppose that Alice computes S = [ h ( M ) ] ~ l i , , and sends M and S to Bob. If Trudy can find a collision with M , that is, if Trudy can find M such that h ( M ) = h ( M ) ,then Trudy can replace M with M , and Bob will erroneously verify the integrity of M . Before considering collision attacks on particular hash functions, we mention one more cryptographic application of hashing. Suppose Alice wants
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H A S H FUNCTIONS
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to send a message M to Bob, and she wants to ensure the integrity of the message. That is, Alice wants Bob to be able to automatically check that the message he receives is the message that was actually sent. Alice has the clever idea that she will compute y = h ( M ) and she will send y and M to Bob. Then Bob can compute the hash of the received message and compare it to y. Bob can thereby verify the integrity of the message, that is, Bob can be confident that he actually received M . There is a serious problem with Alice s integrity scheme. If Trudy intercepts M and y, she can replace M with M , and replace y with y = h(M ). Then when Bob computes the hash of the received message, it will match the received hash value, and he will not suspect that the message has been altered. Seeing the flaw in her scheme, Alice decides instead to compute a keyed hash, that is, y = h ( M ,K ) , where K is a symmetric key that Alice and Bob share and ( M ,K ) denotes the concatenation of M and K . Then, provided that Trudy does not know K , she cannot change the message without the change (almost certainly) being detected. To eliminate some possible attacks, Alice really should compute a socalled HMAC instead of simply appending (or prepending) the key to the message and hashing the result [142]. To understand the potential problem with appending or prepending the key, we need to delve a little deeper into the way that hash functions perform their magic. Most cryptographic hash functions process the message in blocks through several rounds in a manner analogous to the way that block ciphers work. For the two hash functions we consider in this chapter, each round consists of several steps, with each individual step performing a relatively simple operation. The overall hash operation is similar to the way that block cipher CBC mode encryption works (see Section 4.2 for a discussion of CBC mode), since the hash function processes the current block together with the output of the previous block to generate the output for the current block. For the first block, a fixed constant is used in place of the output of the previous block (since there is no previous block), and the output from the last block is the hash value. For the hash functions we consider, the block size is 512 bits, and the hash result is 128 bits. To hash a multi-block message, the 128-bit output of the compression function for block i is added2 to the initial value for block i, and this is used as the initial value when compressing block i 1. The output from the final block is the hash value. Also, the initial value to the first block is denoted as IV, which is a fixed value that is specified as part of the algorithm. The process used to hash multiple blocks is illustrated
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The 128-bit blocks are treated as four 32-bit words and the addition is computed per word. mod1110 z3 .
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