Use RSA public key to generate private key in Openssl?

Question

I know that it is possible using this link to generate a public and a private key for self-signed certificate in OpenSSL. But for a given Public Key, is it possible for me to figure out the corresponding Private Key? I have been using a 1024-bit RSA public key.

Because I had this question in my homework saying:

Generate a digital signature for the sentence “My name is . My voice is my passport.” that verifies correctly using OpenSSL with the following 1024-bit RSA public key. (Hint: The modulus might not have been generated like a normal RSA modulus.):

-----BEGIN PUBLIC KEY-----
MIGdMA0GCSqGSIb3DQEBAQUAA4GLADCBhwKBgQCgF35rHhOWi9+r4n9xM/ejvMEs
Q8h6lams962k4U0WSdfySUevhyI1bd3FRIb5fFqSBt6qPTiiiIw0KXte5dANB6lP
e6HdUPTA/U4xHWi2FB/BfAyPsOlUBfFp6dtkEEcEKt+Z8KTJYJEerRie24y+nsfZ
MnLBst6tsEBfx/U75wIBAw== 
-----END PUBLIC KEY-----

Answer 1

The key (pun intended) to the solution is in the hint:

The modulus might not have been generated like a normal RSA modulus.

This is the approach I took.

Step 1: Derive the primes

1. I first took your file and saved it as public.pem.
2. To get the value of the modulus (n) and the public exponent (e), I ran:
   openssl rsa -pubin -in public.pem -text -noout
3. I converted the hexadecimal modulus and exponent to decimal numbers, which yielded
   n=112420265940019545385580931264662691888876377549063413938338239508058300548918731393322848876821656910452908064089039911552450302375557565600923056341141750687524704844725632296552824986371719004485250857447936962589230504662333990648942759862805127715014382377701044586628936249950092121536791020138692688871
e=3
4. I plugged that number into an online factorialization calculator to discover that one of the two primes used to calculate the modulus was relatively small:
   p=55685342628135644993
q=2018848419246646476894946094575564515176862561629979956283227393349426117194195173357244644821277073710795134539986018769393928719340504755806449531413017314396784334912136112253736003497362080917517151753555605597776865614151048604681116557282512513238254935296910445878892354969335089447

Step 2: Calculate other required values

1. I calculated the totient of n as ϕ(n) = (p - 1) * (q - 1) using the python REPL:
   ϕ(n)=112420265940019545383562082845416045411981431454487849423161376946428320592635503999973422759627461737095663419267762837841655167835571546831529127621801245931718255313312614982156040651459582892231514853950574881671713352908778385051165894248654079110333265820418532073390681314653181675602213322541221954432
2. I then used the python script from this answer to calculate the private exponent (d) and the coefficient (c), which yielded:
   d=74946843960013030255708055230277363607987620969658566282107584630952213728423669333315615173084974491397108946178508558561103445223714364554352751747867497287812170208875076654770693767639721928154343235967049921114475568605852256700777262832436052740222177213612354715593787543102121117068142215027481302955
c=1040291110785843997

Step 3: Create the private key's ASN.1 structure

I then used the calculated values to create an ASN.1 structure in a file named asn as described in this answer:

asn1=SEQUENCE:rsa_key

[rsa_key]
version=INTEGER:0
modulus=INTEGER:112420265940019545385580931264662691888876377549063413938338239508058300548918731393322848876821656910452908064089039911552450302375557565600923056341141750687524704844725632296552824986371719004485250857447936962589230504662333990648942759862805127715014382377701044586628936249950092121536791020138692688871
pubExp=INTEGER:3
privExp=INTEGER:74946843960013030255708055230277363607987620969658566282107584630952213728423669333315615173084974491397108946178508558561103445223714364554352751747867497287812170208875076654770693767639721928154343235967049921114475568605852256700777262832436052740222177213612354715593787543102121117068142215027481302955
p=INTEGER:55685342628135644993
q=INTEGER:2018848419246646476894946094575564515176862561629979956283227393349426117194195173357244644821277073710795134539986018769393928719340504755806449531413017314396784334912136112253736003497362080917517151753555605597776865614151048604681116557282512513238254935296910445878892354969335089447
e1=INTEGER:37123561752090429995
e2=INTEGER:903312890059631
coeff=INTEGER:1040291110785843997

Step 4: Create the private key

Based on the ASN.1 structure, I generated the private key as follows:

1. Create the private key in DER format:
   openssl asn1parse -genconf asn -out private.der
2. Convert the private key to PEM format:
   openssl rsa -in private.der -inform der -out private.pem -outform pem

This results in a private.pem file being created with the following contents:

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

Step 5: Verifying the result

To check whether the created private key (in private.pem) matches the provided public key, I just generated a new public key from the private key:

openssl rsa -in private.pem -pubout

This yields the following output:

writing RSA key
-----BEGIN PUBLIC KEY-----
MIGdMA0GCSqGSIb3DQEBAQUAA4GLADCBhwKBgQCgF35rHhOWi9+r4n9xM/ejvMEs
Q8h6lams962k4U0WSdfySUevhyI1bd3FRIb5fFqSBt6qPTiiiIw0KXte5dANB6lP
e6HdUPTA/U4xHWi2FB/BfAyPsOlUBfFp6dtkEEcEKt+Z8KTJYJEerRie24y+nsfZ
MnLBst6tsEBfx/U75wIBAw==
-----END PUBLIC KEY-----

This output exactly matches the public key that you provided.