The.Algorithm.Design.Manual.Springer-Verlag.1998

Cryptography
● How can I validate the integrity of data against corruption? - In any communications
application, there is need to validate that the transmitted data is identical to that which has been
received. One solution is for the received to transmit the data back to the source and have the
original sender confirm that the two texts are identical. This fails when the exact inverse of an
error is made in the retransmission, but a more serious problem is that your available bandwidth is
cut in half with such a scheme.
A more efficient if less reliable method is to use a checksum, a simple mathematical function that
hashes a long text down to a simple number or digit, and then transmit the checksum along with
the text. The checksum can be recomputed on the receiving end and bells set off if the computed
checksum is not identical to what was received. Perhaps the simplest checksum scheme just adds
up the byte or character values and takes the sum modulo some constant, say .
Unfortunately, an error transposing two or more characters would go undetected under such a
scheme, since addition is commutative.
Cyclic-redundancy check (CRC) provides a more powerful method for computing checksums
and is used in most communications systems and internally in computers to validate disk drive
transfers. These codes compute the remainder in the ratio of two polynomials, the numerator of
which is a function of the input text. The design of these polynomials involves considerable
mathematical sophistication and ensures that all reasonable errors are detected. The details of
efficient computation are complicated enough that we recommend that you start from an existing
implementation, described below.
● How can I prove that a file has not been changed? - If I send you a contract in electronic form,
what is to stop you from editing the file and then claiming that your version was what we had
really agreed to? I need a way to prove that any modification to a document is fraudulent. Digital
signatures are a cryptographic way for me to stamp my document as genuine.
Given a file, I can compute a checksum for it, and then encrypt this checksum using my own
private key. I send you the file and the encrypted checksum. You can now edit the file, but to fool
the judge you must also edit the encrypted checksum such that it can be decrypted to the correct
checksum. With a suitably good checksum function, designing a file that yields the same
checksum becomes an insurmountable problem.
● How can I prove that I am me? - Authentication is the process of one party convincing another
that they are who they say they are. The historical solutions have involved passwords or keys, so I
prove that I am who I say I am because I know my credit card number or carry an ID card. The
problem with such schemes is that anyone who eavesdrops on this conversation or who steals my
physical key can now successfully impersonate me.
What we need is some way for me to convince you that I know my key, without actually telling
you the key. One such method to do so is for you to send me a random number or text, and I use
my key to encrypt your text and send it back to you. If you then decrypt this text and compare it to
the message you sent me, you gain confidence that I am who I say I am. We can repeat this
exercise on several random texts until sufficient authentication has been agreed upon. If we use a
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