Reference no: EM132564203

RSA Public-Key Encryption and Signature Lab

Task 1: Deriving the Private Key

Let p, q, and e be three prime numbers. Let n = p*q. We will use (e, n) as the public key. Please calculate the private key d. The hexadecimal values of p, q, and e are listed in the following. It should be noted that although p and q used in this task are quite large numbers, they are not large enough to be secure. We intentionally make them small for the sake of simplicity. In practice, these numbers should be at least 512 bits long (the one used here are only 128 bits).

Task 2: Encrypting a Message
Let (e, n) be the public key. Please encrypt the message "A top secret!" (the quotations are not included). We need to convert this ASCII string to a hex string, and then convert the hex string to a BIGNUM using the hex-to-bn API BN hex2bn(). The following python command can be used to convert a plain ASCII string to a hex string.

The public keys are listed in the followings (hexadecimal). We also provide the private key d to help you verify your encryption result.

Task 3: Decrypting a Message

The public/private keys used in this task are the same as the ones used in Task 2. Please decrypt the following ciphertext C, and convert it back to a plain ASCII string.

You can use the following python command to convert a hex string back to to a plain ASCII string.

Task 4: Signing a Message

The public/private keys used in this task are the same as the ones used in Task 2. Please generate a signature for the following message (please directly sign this message, instead of signing its hash value):

Please make a slight change to the message M, such as changing $2000 to $3000, and sign the modified message. Compare both signatures and describe what you observe.

Task 5: Verifying a Signature

Bob receives a message M = "Launch a missile." from Alice, with her signature S. We know that Alice's public key is (e, n). Please verify whether the signature is indeed Alice's or not. The public key and signature (hexadecimal) are listed in the following:

Suppose that the signature in is corrupted, such that the last byte of the signature changes from 2F to 3F, i.e, there is only one bit of change. Please repeat this task, and describe what will happen to the verification process.

Task 6: Manually Verifying an X.509 Certificate

In this task, we will manually verify an X.509 certificate using our program. An X.509 contains data about a public key and an issuer's signature on the data. We will download a real X.509 certificate from a web server, get its issuer's public key, and then use this public key to verify the signature on the certificate.

Step 1: Download a certificate from a real web server. We use the www.example.org server in this document. Students should choose a different web server that has a different certificate than the one used in this document (it should be noted that www.example.com share the same certificate with www.example.org). We can download certificates using browsers or use the following command:

The result of the command contains two certificates. The subject field (the entry starting with s:) of the certificate is www.example.org, i.e., this is www.example.org's certificate. The issuer field (the entry starting with i:) provides the issuer's information. The subject field of the second certificate is the same as the issuer field of the first certificate. Basically, the second certificate belongs to an intermediate CA. In this task, we will use CA's certificate to verify a server certificate.
If you only get one certificate back using the above command, that means the certificate you get is signed by a root CA. Root CAs' certificates can be obtained from the Firefox browser installed in our pre-built VM. Go to the Edit ‹ Preferences ‹ Privacy and then Security ‹ View Certificates. Search for the name of the issuer and download its certificate.
Copy and paste each of the certificate (the text between the line containing "Begin CERTIFICATE" and the line containing "END CERTIFICATE", including these two lines) to a file. Let us call the first one c0.pem and the second one c1.pem.

Step 2: Extract the public key (e, n) from the issuer's certificate. Openssl provides commands to extract certain attributes from the x509 certificates. We can extract the value of n using -modulus. There is no specific command to extract e, but we can print out all the fields and can easily find the value of e.

Step 3: Extract the signature from the server's certificate. There is no specific openssl command to extract the signature field. However, we can print out all the fields and then copy and paste the signature block into a file (note: if the signature algorithm used in the certificate is not based on RSA, you can find another certificate).

We need to remove the spaces and colons from the data, so we can get a hex-string that we can feed into our program. The following command commands can achieve this goal. The tr command is a Linux utility tool for string operations. In this case, the -d option is used to delete ":" and "space" from the data.

Step 4: Extract the body of the server's certificate. A Certificate Authority (CA) generates the signature for a server certificate by first computing the hash of the certificate, and then sign the hash. To verify the signature, we also need to generate the hash from a certificate. Since the hash is generated before the signature is computed, we need to exclude the signature block of a certificate when computing the hash. Finding out what part of the certificate is used to generate the hash is quite challenging without a good understanding of the format of the certificate.

X.509 certificates are encoded using the ASN.1 (Abstract Syntax Notation.One) standard, so if we can parse the ASN.1 structure, we can easily extract any field from a certificate. Openssl has a command called asn1parse, which can be used to parse a X.509 certificate.

The field starting from 0 is the body of the certificate that is used to generate the hash; the field starting from A is the signature block. Their offsets are the numbers at the beginning of the lines. In our case, the certificate body is from offset 4 to 1249, while the signature block is from 1250 to the end of the file. For
X.509 certificates, the starting offset is always the same (i.e., 4), but the end depends on the content length of a certificate. We can use the -strparse option to get the field from the offset 4, which will give us the body of the certificate, excluding the signature block.
$ openssl asn1parse -i -in c0.pem -strparse 4 -out c0_body.bin -noout
Once we get the body of the certificate, we can calculate its hash using the following command:
$ sha256sum c0_body.bin

Step 5: Verify the signature. Now we have all the information, including the CA's public key, the CA's signature, and the body of the server's certificate. We can run our own program to verify whether the signature is valid or not. Openssl does provide a command to verify the certificate for us, but students are required to use their own programs to do so, otherwise, they get zero credit for this task.

Attachment:- RSA Public-Key Encryption and Signature Lab.rar