How to Generate rsa keys using specific input numbers in openssl?

if you are looking for a method that requires no other computations from you, than I guess such do not exist.

have a look at rsa_builtin_keygen in rsa_gen.c of openssl source, it generates the primes for itself.

I would suggest you have a look at function generateKey here (you will also need the file cryptomath for it to work)

One way to do this is to generate a DER encoded key using OpenSSL's asn1parse command's -genconf option.

You'll need to construct an input file for asn1parse -genconf to produce an RSA key in the standard format (per RFC 3447). The syntax for asn1parse -genconf is given here: http://www.openssl.org/docs/crypto/ASN1_generate_nconf.html and indeed, it already has an example for constructing an RSA key.

You need to calculate a few more values (specifically, d mod (p-1), d mod (q-1) and q^-1 mod p. For the values of p, q, d you gave, these are:

d mod(p-1) = 23 mod 16 = 7

d mod(q-1) = 23 mod 10 = 3

q^-1 mod p = 14

Put this all together into a text file in the appropriate format:

asn1=SEQUENCE:rsa_key

[rsa_key]
version=INTEGER:0
modulus=INTEGER:187
pubExp=INTEGER:7
privExp=INTEGER:23
p=INTEGER:17
q=INTEGER:11
e1=INTEGER:7
e2=INTEGER:3
coeff=INTEGER:14

To construct the binary DER file:

openssl asn1parse -genconf <path to above file> -out newkey.der

You can then run this through OpenSSL's rsa command to confirm:

openssl rsa -in newkey.der -inform der -text -check

Which should output:

Private-Key: (8 bit)
modulus: 187 (0xbb)
publicExponent: 7 (0x7)
privateExponent: 23 (0x17)
prime1: 17 (0x11)
prime2: 11 (0xb)
exponent1: 7 (0x7)
exponent2: 3 (0x3)
coefficient: 14 (0xe)
RSA key ok
writing RSA key
-----BEGIN RSA PRIVATE KEY-----
MBwCAQACAgC7AgEHAgEXAgERAgELAgEHAgEDAgEO
-----END RSA PRIVATE KEY-----

You can use this to encrypt data with OpenSSL's rsautl command (though with this key you're limited to encrypting just a single byte of data providing that byte is also less than 187).