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can be used t o attack NTRU. The numbers t and s are chosen so that the average number of collisions between
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When this number is near one, appropriate values for t and s have been deterrnined for the chosen-ciphertext attack. In [73],the authors present a n example of complete key recovery, using the highest security set of parameters described in [138]. They also provide estimated running times for different sets of parameters using different, values o f t , s , and y. Although we will not discuss it here, Jaiilmes and Joux also shows that this attack ca,n be modified so that it is effective against OAEP-like padding within NTRU. One way to thwart, this chosen-ciphertext attack is t o use othcr padding methods as described in [54].
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Like any respectable cryptosysteni, NTRU has been under close scrutiny sirice its invention. As attacks arid weaknesses have been discovered, the implementation-as well as the recommended security parameters N , p , q , L j . L,, and L,r--have evolved over time. In fact, NTRU encryption is now in its third major revision, due t o viable attacks on earlier v e r ~ i o n s . ~ Of course, Kerckhoffs Principle dictates that a cryptosystem must be subject t o extensive investigation. However, the evolution of NTRU is in stark contrast to, say, RSA, which has undergone no significant revisions since its invention. Given these track records, it could be argued that RSA likely rests on a sounder foundat,ion than NTRU. Nevertheless, there are no known weaknesses in the current version of NTRU encryption [ l o l l . In any case, it appears t,hat NTRU is a crypt,osystt:m with a future, in contrast to many other proposed public key systems-although a cynic might argue that this has as much t o do with patents and the heavy corporate backing NTRU has received than with any inherent technical superiority. Tirnc will tell whether the current version of NTRU proves more durable than its predecessors.
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Drink nothing without seeing it; sign nothing without reading it. - Spanish Proverb
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Public key cryptography can be used to create digital signatures. If properly implemented, when Bob receives a message digitally signed by Alice, he is assured that it was composed by Alice--assuming that Alice s private key is private. Several different digital signature schemes have been proposed. For example, RSA can be used for signing by simply using the private key to encrypt and the public key to verify the signature. ElGamal is another signature scheme, which we discuss in this section. In this section, we employ the public key notation defined previously in Section 5.1. That is, we use the following notation:
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Encrypt message M with Alice s public key: C = { M } ~ l i ~ ~ . Decrypt ciphertext
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c with Alice s private key: M = [C]Alice.
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Signing and decrypting are the same operations, so the notation for Alice signing message M is S = [M]*lice, where S is the signed message.
Encryption and decryption are inverse operations so that
[{M)AlicelAlice = {[MIAlice)Alice = M .
It is important to remember that only Alice can sign since the signature requires Alice s private key. However, anyone can verify Alice s signature, since that is a public key operation. We can define confidentiality as no unauthorized reading and integrity as no unauthorized writing [142]. By using Bob s public key, Alice can send encrypted messages to Bob and be assured of confidentiality. For integrity, Alice can use a digital signature, as discussed in Section 5.1. Both integrity and confidentiality can be achieved by using symmetric key cryptography, but a digital signature also provides non-repudiation, which cannot be achieved with symmetric keys. How can Alice have confidentiality, integrity and non-repudiation using a public key cryptosystem There are two natural strategies which Alice might use to accomplish this, that is, she can sign the message M and then encrypt the result before sending it to Bob or she can encrypt M and then sign the result before sending it to Bob. Using scenarios found in [142], we will show that both of these strategies have potential pitfalls. First, suppose that Alice and Bob are romantically