Problem: Calculate the integral

Feynman’s method: Solve the following equation for $t$

and calculate $I(0)$.

These are my notes on the book Cryptography Engineering by
Niels Ferguson, Bruce Schneier, Tadayoshi Kohno
Paperback: 384 pages
Publisher: John Wiley & Sons; March 2010
ISBN: 978-0470474242

“The world is full of bad security systems designed by people who have read Applied Cryptography.” Cryptography Engineering could have the same effect.

Randomness

Generate 33 random bytes (wrapped in 44 characters)

openssl rand -base64 33 -out pass.txt

Output random bytes in hex format

openssl rand -hex 33

Certificate Authority

Create a CA certificate (refer to openssl-ca.cnf file)

openssl req -x509 -config openssl-ca.cnf -newkey rsa:4096 -sha256 -days 3000 -out cacert.pem -keyout cakey.pem -passout file:pass.txt

RFC8017 (PKCS #1 v.2.2)

Defines the traditional format for RSA keys. Two structures:

-----BEGIN RSA PRIVATE KEY-----
RSAPrivateKey
-----END RSA PRIVATE KEY-----

and

-----BEGIN RSA PUBLIC KEY-----
RSAPublicKey
-----END RSA PUBLIC KEY-----

Commands

Generate RSA private key

openssl genrsa -out private.pem 2048

Extract public key from RSA private key

openssl rsa -in private.pem -out public.pem -RSAPublicKey_out

“What I cannot create, I do not understand” — Richard Feynman

As a fun exercise I wanted to know how to build an RSA private key file from scratch. It turned out that it was not complicated. In fact it was very educational to learn about the PKCS #1 and X.609 standards. By the end of this post you should be able to read DER files without the openssl command, too.

From a mathematical perspective an RSA private key is just a pair of numbers satisfying a few conditions. The first step is to choose two prime numbers. If we want our key to be $k$-bit long then each prime should be $k/2$-bit long. Here are two primes 1024-bit long:

p = 188658351657909995564241240465674883756750911978965594406091265432265641964092435440867010496656290185915042088848864982944624560228571535282140345941898374783486396513112284924130530433476612244870421527834092351771495657957917171265855063528745445693207773620468819387929613829761428627329588653924255089451
q = 314178598271171309643469864042809599136290738797354680579453738873053976803051841787567944451717551290169456210178094252448060684978774667964812128639960504040754248179131205124174037497556941333232996617690501240860884613659741212508891734662698093999296965839284246675187499601986034176733136066305345935229