Cryptography in Cyber Security

The factorization of an arbitrarily large number N, into its constituent primes, determining the powers n2, n3, n5, n7, etc. of the primes, is computationally infeasible. As far as we know.

Decryption uses factorization internally. So, it is computationally infeasible. Note that this is assuming that we are using known methods, including brute force.

Is it possible that some one or some country has actually discovered fast algorithms, but chose to keep them secret, for these tasks that we believe to be computationally infeasible?

This is not the same thing as saying that computational infeasibility is the same idea as Turing-incomputable.

Nor is it the same thing as saying that you cannot make a lucky guess, or heuristically arrive at a possible answer, and then systematically verify that the guessed answer is indeed the correct answer, all done within a matter of seconds on a lowly PC.

Here is an example: Microsoft's old Windows NT used the DES encryption algorithm in storing the passwords. Brute-forcing such a scrambled password to compute the plain text password can take, according to Microsoft, "about a billion years." But https://www.l0pht.com claims that L0phtCrack breaks Windows passwords in about one week, running in the background on an old Pentium PC.

1. As of 2018: Most popular public-key algorithms can be efficiently broken by a sufficiently strong hypothetical quantum computer.
2. https://en.wikipedia.org/wiki/Post-quantum_cryptography Recommended Reading.

1. One-way hash functions are also known as message digests (MD), fingerprints, or compression functions.

1. The most popular one-way hash algorithms are MD4 and MD5 (both producing a 128-bit hash value), and SHA Secure Hash Algorithm, also known as SHA1 (producing a 160-bit hash value).
2. As of 2006, both MD5 and SHA1 are considered separately broken. That is, given plain text p, it is possible to modify p to a desired p' so that md5(p) = md5(p').
3. Similarly, for SHA1. What is not known is if we can modify p to a p' so that md5(p) = md5(p') and sha1(p) = sha1(p').

1. SHA2, a successor to SHA1, is a range of hash functions and includes closely related SHA224, SHA256 (256 bits long), SHA384 and SHA512 (512 bits long). SHA3 is released by NIST in 2015. https://en.wikipedia.org/wiki/SHA-3.

1. Read the man pages: man md5 and man md5sum and man sha1sum and man hashalot etc.
   1. % sha512sum kubuntu-cosmic-desktop-amd64.iso # stdout shown folded

      851238208f71114e64dd3dfd2bd516d097823fb5fe94432a7ed4dfad02dfe8f1
      ce966f9f04793ce019869ee241db9044ade7219fcff8fe73db406c4a4a3b94f0
      kubuntu-cosmic-desktop-amd64.iso
      

   2. % sha256sum kubuntu-cosmic-desktop-amd64.iso

      3f470978690b8fb343c94b8a8ded62f0372f9837ced583eba84045480d79d065
      kubuntu-cosmic-desktop-amd64.iso
      

   3. % md5sum kubuntu-cosmic-desktop-amd64.iso

      bb93c40531b7b13fe7b01a9c75bbc312  kubuntu-cosmic-desktop-amd64.iso
      

A popular symmetric-key system is the DES, short for Data Encryption Standard. DES was developed in 1975 and standardized by ANSI in 1981. DES encrypts data in 64-bit blocks using a 56-bit key. The algorithm transforms the input in a series of steps into a 64-bit output.

IDEA (International Data Encryption Algorithm) is a block cipher which uses a 128-bit length key to encrypt successive 64-bit blocks of plain text. The procedure is quite complicated using subkeys generated from the key to carry out a series of modular arithmetic and XOR operations on segments of the 64-bit plaintext block. The encryption scheme uses a total of fifty-two 16-bit subkeys.

Blowfish is a symmetric block cipher that can be used as a drop-in replacement for DES or IDEA. It takes a variable-length key, from 32 bits to 448 bits, making it ideal for both domestic and exportable use. Blowfish is unpatented and license-free, and is available free for all uses.

The most well-known of the public-key encryption algorithms is RSA, named after its designers Rivest, Shamir, and Adelman. The un-breakability of the algorithm is based on the fact that there is no efficient way to factor very large numbers into their primes.

1. Find two (large) primes, p and q.
2. Compute the product, n = p*q (called, the public modulus).
3. Choose e (the public exponent), such that
   1. e < n, and
   2. e is relatively prime to (p-1)*(q-1)
4. Compute d (the private exponent) such that (e*d) mod (p-1)*(q-1) = 1.
5. Public-key is (n, e), and the private key is (n, d).

An example of the above numbers: ./rsa.txt. Look up the man page: openssl(1).

The/e/ and d are symmetric in that using either ((n,e) or (n,d)) as the encryption key, the other can be used as the decryption key.

The only way known to find d is to know p and q. If the number n is small, p and q are easy to discover by prime factorization. Thus, p and q are chosen to be as large as possible, say, a few hundred digits long. Obviously, p and q should never be revealed.

The Digital Signature Algorithm (DSA) is a United States Federal Government standard for digital signatures.

1. Choose a prime q. Choose a prime modulus p such that p - 1 is a

multiple of q.

1. Choose g, a number whose multiplicative order modulo p is q. (This may be done by setting g = h^{((p - 1)/q)} mod p for some arbitrary h (1 < h < p-1), and trying again with a different h if the result comes out as 1. Most choices of h will lead to a usable g; commonly h=2 is used. )1. Choose x by some random method, where 0 < x < q.
2. Calculate y = g^x mod p.
3. Public key is (p, q, g, y), and the private key is x.

An example of the above numbers: ./dsa.txt Look up the man page: openssl(1).

Public-key systems, such as Pretty Good Privacy (PGP), are popular for transmitting information via the Internet. They are extremely secure and relatively simple to use. You need to retrieve the recipient's public key from one of several world-wide registries of public keys that now exist to encrypt a message.

When John wants to send a secure message to Jane, he uses Jane's public key to encrypt the message. Jane then uses her private key to decrypt it.

In real-world implementations, public keys are rarely used to encrypt actual messages because public-key cryptography is slow. Instead, public-key cryptography is used to distribute symmetric keys, which are then used to encrypt and decrypt actual messages, as follows:

1. Bob sends Alice his public key. (Or, Alice retrieves Bob's public key from a registry.)
2. Alice generates a (random) symmetric key (called a session key), encrypts it with Bob's public key, and sends it to Bob.
3. Bob decrypts the session key with his private key.
4. Alice and Bob exchange messages using the session key.

An independent third party that everyone trusts, whose responsibility is to issue certificates, is called a Certification Authority (CA).

A package containing a person's name or website's name … (and possibly some other information such as an E-mail address and company name) and his public key and signed by a trusted third party is called a digital certificate (or digital ID). A digital certificate certifies the ownership of a public key by the named subject of the certificate.

A digital certificate serves two purposes. First, it provides a cryptographic key that allows another party to encrypt information for the certificate's owner. Second, it provides a measure of proof that the holder of the certificate is who they claim to be.

The recipient of an encrypted message uses (i) the CA's public key to decode the digital certificate attached to the message, (ii) verifies the certificate as issued by the CA, (iii) obtains the sender's public key and identification information held within the certificate. The message must have been encrypted with the private key of the sender.

The most widely used standard for digital certificates is X.509. What are colloquially known as SSL certificates are X.509 certificates. X.509 uses public key infrastructure (PKI) standard. It defines the following:

1. Version field identifies the certificate format.
2. Serial Number unique within the CA.
3. Signature Algorithm identifies the algorithm used to sign the certificate.
4. Issuer Name is the name of the CA.
5. Period of Validity is a pair of Not Before, and Not After Dates
6. Subject Name is the name of the user to whom the certificate is issued
7. Subject's Public Key field includes algorithm name and the public key of the subject.
8. Extensions
9. Signature of CA.

1. "You can either buy an SSL (X.509) certificate or generate your own (a self-signed certificate) for testing or, depending on the application, even in a production environment. Good news: If you self-sign your certificates you may save a ton of money. Bad news: If you self-sign your certificates nobody but you and your close family (perhaps) may trust them."
2. https://letsencrypt.org/ Let’s Encrypt is a free, automated, and open Certificate Authority.
3. IRS approved certificate authorities: https://www.irs.gov/businesses/corporations/digital-certificates

1. The digital signature is associated with a document to authenticate where the document originated.
   1. Alice the sender.
      1. Compute a hash H of the document D.
      2. Encrypt H with the sender’s private key. Result is E.
      3. Send the E and the digital certificate DCA of Alice.
   2. Bob the recipient of the document D, E, and DCA.
      1. Compute a hash H' of D.
      2. Decrypts E, the signature, with Alice's public key from DCA. Result is E'.
      3. Compare the two values, E' == H'?. If they match, Bob knows that:
         1. the document really came from Alice and
         2. the document was not tampered with during transmission.
2. A digital certificate contains the digital signature of the certificate-issuing authority.
3. "While digital signature is a technical term, defining the result of a cryptographic process that can be used to authenticate a sequence of data, the term electronic signature – or e-signature – is a legal term that is defined legislatively."