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I need to sign the numbers from 1 to 1 billion (literally this, it's not an analogy) using a digital signature and then send these numbers in a particular order to someone. The message is not private and the receiver should be able to verify its authenticity without having to reply; it's a one-way public communication channel.

Are there any algorithms that are safe enough for this? How many bits should the key have and how many bits will the signature have?

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You might want to pay close attention to exactly what "authentic" means in this context. It just means that the sender at some point in time, indeed signed a message with particular contents. Hence, replay attacks might be a concern in similar scenarios. If you need to know when and in response to what request the sender signed the message, the sender will have to sign that information too. –  Henrick Hellström May 7 '13 at 9:37

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Yes. Any good standard digital signature algorithm will be secure in this setting.

Digital signature algorithms are designed to be secure against chosen-message attacks, where the attacker can choose any set of messages and learn the signatures on those messages; the security of the signature scheme means that this doesn't help the attacker at all. This means that it's safe to sign arbitrary messages; there's nothing dangerous or special about messages that are small numbers.

You should choose the key size based upon the desired security level. A bare-minimum security level is 80-bit security; that then translates to a key size (in a way that is dependent upon the specific signature algorithm you choose). Similarly, the size of the signature will depend upon the particular signature algorithm you use.

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Perhaps make that any good digital signature algorithm; some standards have been lacking in this respect; e.g; ISO/IEC 9796 (better known as ISO/IEC 9796-1) which was withdrawn following attacks, and ISO/IEC 9796-2 which has a more or less serious vulnerability in its original (and most used) mode. –  fgrieu May 7 '13 at 10:35

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