Your problem is located in the file format you are actually using with Java. You probably save the
private and the public in encoded ("byte array") to a file and rebuild the keys e.g. with
X509EncodedKeySpec.
This format is not compatible to Node.JS and you have 3 ways to solve it:
a) you write the keys in Java with re neccessary format for usage in Node.JS
b) you write a converter in Node.JS to get the correct format
c) you convert the files with a tool like OPENSSL.
Here I show you the "c-way" as you are handling just one keypair and probably don't need a programatically solution.
Let's say you have two files with the private key ("rsa_privatekey_2048.der") and the public key ("rsa_publickey_2048.der").
In OPENSSL you are using the command line with
openssl rsa -inform der -in rsa_privatekey_2048.der -outform pem -out rsa_privatekey_2048.pem
openssl rsa -inform der -pubin -in rsa_publickey_2048.der -outform pem -RSAPublicKey_out -out rsa_publickey_2048.pem
to convert the files to their PEM-encoded formats.
Below you can find the two sample files I created.
rsa_privatekey_2048.pem:
-----BEGIN RSA PRIVATE KEY-----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-----END RSA PRIVATE KEY-----
rsa_publickey_2048.pem:
-----BEGIN RSA PUBLIC KEY-----
MIIBCgKCAQEAmbeKgSAwVe0nZ84XlbDhMkUDjx1C0duA16MkzHTg1uh9SouOKK0e
3gPtTJ9LssaHlXSYhjpMDMWGO6ujd85XRosI2u9eSMNRYY25AQuBriSTVdi9BHqW
AuWuo6VuvTrkgWTL69vNWvLXTOkTiIyrgnhiavjNvm4UVy2AcO2Y3ER+dKgJpQAY
lEP1jvuQuf6dfNdSBoN0DZbxZXYbQqoA9R/u0GZHCXY+r8A54RejG34pnnuHkoyR
OZz5H9LbKGOiaETryornQ1TRvB/p9tgIoCJFI71WsKsqeWQPG3Ymg/FoEWXNY0yo
pZEjpkZa3tU+hrOmAFIRg+/bedKfjYFi/QIDAQAB
-----END RSA PUBLIC KEY-----