*N*be a product of two, very large, primes

*p*and

*q*.

*N*is n-bit number, those days minimally 1024 bits, and thus

*p*and

*q*are half that size, so that after multiplication we get required number of bits. Next, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365re are two numbers

*e*and

*d*that satisfy relation , where . is encryption key - or what's frequently referred to as a public key, while is decryption key, or a private key. The question now is: Given a PEM file with private key, how to find out those parameters for a specific public/private keys? In this post I'll show three ways to do it. The first two are using tools that are a standard part of OpenSSL/GnuTLS libraries. The third one is using Python.

**OpenSSL/GnuTLS**

Those two, especially OpenSSL, are very popular and complex cryptograhpic libraries. Both of cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365m come with a very capable command line tools. In cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 case of OpenSSL cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365re is a single binary, openssl, that performs its function depending on cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 second argument. In this post, I'll use arguments rsa and genrsa that manipulate and generate, respectively, RSA keys. GnuTLS, on cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 ocá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365r hand, has a several tools, of which we'll use certtool.

Now, first thing is to create RSA public/private key. You can do this using cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following commands:

openssl genrsa -out key.pem 512or, if you use GnuTLS, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365n:

certtool --generate-privkey --bits 512 --outfile key.pemIn both cases cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 generated RSA modulus (N) will have 512 bits, and cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 keys will be written into output file key.pem. You should note few things here:

- The output file contains both private and public keys!
- We didn't encrypt output file, which is recommended to do.
- 512 bits cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365se days is a way insecure. Use minimally 1024 bits.

openssl rsa -in key.pem -noout -textor, using GnuTLS:

certtool -k --infile key.pemIn both cases you'll receive cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following output (GnuTLS is a bit more verbose):

Private-Key: (512 bit)In this output we see parameters

modulus:

00:ba:b6:78:3b:1c:15:f1:d9:e3:48:16:5e:e7:8e:

fd:a0:9d:2f:ee:1b:b8:9b:3d:d3:ea:f4:ad:fb:1b:

6e:ef:b2:b5:cd:ee:38:e9:f8:6d:64:c9:ea:95:ae:

87:13:5a:23:8b:2f:0b:e8:bb:c6:f8:c6:c4:ee:64:

3c:d4:97:bd:a3

publicExponent: 65537 (0x10001)

privateExponent:

39:6b:07:ca:55:b6:c1:eb:59:a3:bf:8d:6b:f4:63:

36:d3:5f:fb:ff:76:63:f7:3d:86:51:bc:77:2e:56:

8d:4b:87:73:e0:53:bd:17:e8:4a:e8:df:f5:86:14:

65:60:f2:4f:03:02:3b:e9:23:c6:d3:ce:b3:1d:e9:

13:1a:0f:b1

prime1:

00:d6:b1:f9:53:8c:56:96:79:c0:bd:68:6c:b9:07:

e7:9c:70:de:f5:61:ed:bb:51:12:1d:24:37:0f:cc:

bf:8a:95

prime2:

00:de:a2:52:be:a1:a4:eb:d7:48:24:95:c5:2c:05:

bd:5f:7f:74:d5:12:bd:7c:5f:f1:8e:45:a2:50:26:

ec:d1:57

exponent1:

66:61:30:58:1b:10:1f:69:a7:f3:aa:9c:4e:0f:ea:

ee:bb:14:57:47:7f:aa:57:9a:9f:b2:e9:5e:eb:70:

5b:91

exponent2:

22:2d:8f:40:5e:b6:5f:d2:5b:eb:e9:e6:2c:1c:f1:

76:90:ad:91:ec:5f:94:91:72:16:e2:4f:c9:b8:40:

10:df

coefficient:

22:9c:f3:1f:85:68:a3:36:ab:07:87:ed:a4:c0:e5:

ef:13:a8:28:02:55:35:c1:76:96:86:97:58:08:90:

6e:70

*N*(

*modulus*),

*e*(

*publicExponent*),

*d*(

*privateExponent*),

*p*(

*prime1*), and

*q*(

*prime2*). Also, what's written in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365re are values d mod (p-1) (

*exponent1*), d mod (q-1) (

*exponent2*) and q

^{-1}mod p (

*coefficient*).

The interesting thing to note is cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 value of

*publicExponent*, it is 65537. Namely, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 size of public exponent isn't so important and by having such a low value it is relatively easy to encrypt messages (rising to this exponent requires 17 multiplications). This is very frequent value for encryption key, i.e. public exponent.

*privateExponent*, on cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 ocá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365r hand, has to be large and random, so that it isn't easy to guess.

**Python**

The recommended library to use to manipulate RSA keys is m2crypto which is a wrapper to OpenSSL library. There are many ocá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365r libraries, of course, and you can find an older comparison here. Unfortunately, I was unable to find downloadable version on cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 Internet.

Anyway, to be able to manipulate RSA keys you have to import RSA module from M2Crypto, i.e.:

from M2Crypto import RSAThen, to load RSA private/public key from a file use cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following line:

rsa=RSA.load_key('key.pem')You can also generate new RSA private key as follows:

rsa=RSA.gen_key(512, 65537)In this case you are generating private key based on public key 65537 (we saw that this is very frequently used public key) and we require 512 bit modulus. You can find out modulus length using Python's len() function, i.e.:

>>> len(rsa)To obtain N, you can inquire attribute n of cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 object holding RSA key, i.e.:

512

>>> rsa.nTo obtain pair (e,d), or public key, you can use cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following method:

>>> RSA.pub()Alternatively, you can find out only e in cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365 following way:

>>> rsa.eAs far as I could see, cá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365re is no way to find ocá cược thể thao bet365_cách nạp tiền vào bet365_ đăng ký bet365r parameters besides N, e and bit length.

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