# Logbook for Lab 05
## Purpose
This labs purpose is to improve your understanding of the RSA algorithm by going through all the essential steps of the RSA algorithm in real numbers, applying the learned theories.
## Task 1: Deriving the Private Key
To calculate the private key, this program was created in C:
```c=
#include <stdio.h>
#include <openssl/bn.h>
#define NBITS 128
void printBN2(char *msg, BIGNUM *a, BIGNUM *b)
{
char *number_str_a = BN_bn2hex(a);
char *number_str_b = BN_bn2hex(b);
printf("%s (%s,%s)\n", msg, number_str_a, number_str_b);
OPENSSL_free(number_str_a);
OPENSSL_free(number_str_b);
}
void printBN(char *msg, BIGNUM *a)
{
char *number_str_a = BN_bn2hex(a);
printf("%s %s\n", msg, number_str_a);
OPENSSL_free(number_str_a);
}
int main(){
// init
BN_CTX *ctx = BN_CTX_new();
BIGNUM *p = BN_new();
BIGNUM *q = BN_new();
BIGNUM *n = BN_new();
BIGNUM *e = BN_new();
BIGNUM *p_1 = BN_new();
BIGNUM *q_1 = BN_new();
BIGNUM *phi = BN_new();
BIGNUM *prime = BN_new();
BIGNUM *d = BN_new();
// assign value
BN_hex2bn(&p, "F7E75FDC469067FFDC4E847C51F452DF");
BN_hex2bn(&q, "E85CED54AF57E53E092113E62F436F4F");
BN_hex2bn(&e, "0D88C3");
// Public key n = pq
BN_mul(n, p, q, ctx);
printBN2("Public Key is ", e, n);
// phi(n) = (p-1)(q-1)
BN_sub(p_1, p, BN_value_one());
BN_sub(q_1, q, BN_value_one());
BN_mul(phi, p_1, q_1, ctx);
// check if "e" and phi is relatively prime
BN_gcd(prime, phi, e, ctx);
if (!BN_is_one(prime))
{
printf("Error: e and phi(n) is not relatively prime! \n ");
exit(0);
}
// Calculate Private key
BN_mod_inverse(d, e, phi, ctx);
printBN("Private Key is ", d);
}
```
After running the created program, we managed to get the private key.
## Task 2: Encrypting a Message
We need to convert this ASCII string ("A top secret!") to a hex string, and then convert the hex string to a BIGNUM using the hex-to-bn API BN_hex2bn().
Converting the string to hexadecimal:
```
python -c 'print("A top secret!".encode("hex"))'
4120746f702073656372657421
```
To encrypt the message we created the following code:
```c=
#include <stdio.h>
#include <openssl/bn.h>
void printBN(char *msg, BIGNUM *a)
{
char *number_str_a = BN_bn2hex(a);
printf("%s %s\n", msg, number_str_a);
OPENSSL_free(number_str_a);
}
int main()
{
BN_CTX *ctx = BN_CTX_new();
BIGNUM *n = BN_new();
BIGNUM *e = BN_new();
BIGNUM *M = BN_new();
//encrypt variable
BIGNUM *C = BN_new();
// provided values
BN_hex2bn(&n, "DCBFFE3E51F62E09CE7032E2677A78946A849DC4CDDE3A4D0CB81629242FB1A5");
BN_dec2bn(&e, "65537"); //decimal
// message "A top secret!"
BN_hex2bn(&M, "4120746f702073656372657421");
// encrypt message M: M^e mod n
BN_mod_exp(C, M, e, n, ctx);
printBN("Encryption message:", C);
return 0;
}
```
By running the program we get the encrypted message:
## Task 3: Decrypting a Message
Using the same public and private keys as in task 2, let's now decrypt the ciphertext C, and convert it back to a plain ASCII string.
`C = 8C0F971DF2F3672B28811407E2DABBE1DA0FEBBBDFC7DCB67396567EA1E2493F`
To decrypt the ciphertext C and get the hexadecimal string, we create the following program:
```c=
#include <stdio.h>
#include <openssl/bn.h>
void printBN(char *msg, BIGNUM *a)
{
char *number_str_a = BN_bn2hex(a);
printf("%s %s\n", msg, number_str_a);
OPENSSL_free(number_str_a);
}
int main()
{
BN_CTX *ctx = BN_CTX_new();
BIGNUM *n = BN_new();
BIGNUM *M = BN_new();
BIGNUM *d = BN_new();
BIGNUM *C = BN_new();
// provided values
BN_hex2bn(&n, "DCBFFE3E51F62E09CE7032E2677A78946A849DC4CDDE3A4D0CB81629242FB1A5");
BN_hex2bn(&d, "74D806F9F3A62BAE331FFE3F0A68AFE35B3D2E4794148AACBC26AA381CD7D30D");
// message crypt
BN_hex2bn(&C, "8C0F971DF2F3672B28811407E2DABBE1DA0FEBBBDFC7DCB67396567EA1E2493F");
// decrypt C: C^d mod n
BN_mod_exp(M, C, d, n, ctx);
printBN("Decryption message:", M);
return 0;
}
```
Thus obtaining the hexadecimal string:
Now converting the hexadecimal string back to a simple ASCII string.
## Task 4: Signing a Message
In order to sign a message we first need to calculate the cipher with the algorithm used in the previous task.
We calculate the hexadecimal values for the messages using:
```
python -c 'print("I owe you $2000".encode('utf-8').hex())'
49206f776520796f75202432303030
```
The hexadecimal value for the message "I owe you $2000" is `49206f776520796f75202432303030`
And,
```
python -c 'print("I owe you $3000".encode('utf-8').hex())'
49206f776520796f75202433303030
```
The hexadecimal value for the message "I owe you $3000" is `49206f776520796f75202433303030`
With this values we calculate the signature for each message using this code
```c=
BN_CTX *ctx = BN_CTX_new();
BIGNUM *n = BN_new();
BIGNUM *e = BN_new();
BIGNUM *M = BN_new();
BIGNUM *d = BN_new();
BIGNUM *c = BN_new();
BN_hex2bn(&n, "DCBFFE3E51F62E09CE7032E2677A78946A849DC4CDDE3A4D0CB81629242FB1A5");
BN_hex2bn(&e, "010001");
BN_hex2bn(&M, "4120746f702073656372657421");
BN_hex2bn(&d, "74D806F9F3A62BAE331FFE3F0A68AFE35B3D2E4794148AACBC26AA381CD7D30D");
BIGNUM *signature = BN_new();
// I owe you $2000
BN_hex2bn(&M, "49206f776520796f75202432303030");
BN_mod_exp(signature, M ,d , n, ctx);
printBN("Signature for $2000 = ", signature);
printf("\n");
//I owe you $3000
BN_hex2bn(&M, "49206f776520796f75202433303030");
BN_mod_exp(signature, M ,d , n, ctx);
printBN("Signature for $3000 = ", signature);
printf("\n");
```
Signatures:
## Task 5: Verifying a Signature
To verify the signature of the message we use inverse operation of the method used in the previous task:
```c=
BIGNUM *S = BN_new();
/* M = Launch a missile. Hex = 4c61756e63682061206d697373696c652e
S = 643D6F34902D9C7EC90CB0B2BCA36C47FA37165C0005CAB026C0542CBDB6802F
e = 010001 (this hex value equals to decimal 65537)
n = AE1CD4DC432798D933779FBD46C6E1247F0CF1233595113AA51B450F18116115
*/
BN_hex2bn(&n, "AE1CD4DC432798D933779FBD46C6E1247F0CF1233595113AA51B450F18116115");
BN_hex2bn(&e, "010001");
BN_hex2bn(&M, "4c61756e63682061206d697373696c652e");
BN_hex2bn(&S, "643D6F34902D9C7EC90CB0B2BCA36C47FA37165C0005CAB026C0542CBDB6802F");
BN_mod_exp(c, S, e, n, ctx);
printBN("M given = ", M);
printBN("M obtained =", c);
```
Using the code above we obtain a M equal to the one provided:
```
M given = 4C61756E63682061206D697373696C652E
M obtained = 4C61756E63682061206D697373696C652E
```
Simulating an error in the Signature by changing the last 2 bytes from `2F` to `3F`.
```c=
BN_hex2bn(&S, "643D6F34902D9C7EC90CB0B2BCA36C47FA37165C0005CAB026C0542CBDB6803F");
BN_mod_exp(c, S, e, n, ctx);
printBN("M given = ", M);
printBN("M obtained =", c);
```
```
M given = 4C61756E63682061206D697373696C652E
M obtained = 91471927C80DF1E42C154FB4638CE8BC726D3D66C83A4EB6B7BE0203B41AC294
```
Has we can see the M obtained is cleary different than the one provided. We can conclude that even a single byte change in a signature produces massive difference in the decrypted message obtained.
## Task 6: Manually Verifying an X.509 Certificate
### Step 1: Download a certificate from a real web server
For this step we downloaded a certificate from `www.santander.pt:443` using the following command:
```bash
openssl s_client -connect www.santander.pt:443 -showcerts > santander.txt
```
This gave us the following text:
```
CONNECTED(00000003)
---
Certificate chain
0 s:C = ES, ST = Cantabria, L = Santander, jurisdictionC = ES, O = "Grupo Santander (Banco Santander, S.A.)", businessCategory = Private Organization, OU = TOTTA, serialNumber = A39000013, CN = www.santander.pt
i:C = US, O = "Entrust, Inc.", OU = See www.entrust.net/legal-terms, OU = "(c) 2014 Entrust, Inc. - for authorized use only", CN = Entrust Certification Authority - L1M
-----BEGIN CERTIFICATE-----
MIIHZjCCBk6gAwIBAgIQUPHwvnH98PnjkgjHvl+XRjANBgkqhkiG9w0BAQsFADCB
ujELMAkGA1UEBhMCVVMxFjAUBgNVBAoTDUVudHJ1c3QsIEluYy4xKDAmBgNVBAsT
H1NlZSB3d3cuZW50cnVzdC5uZXQvbGVnYWwtdGVybXMxOTA3BgNVBAsTMChjKSAy
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mO3tTXEkIuy626KbagsnpVTO6krhhlQWNKVFdRnCWIl7B02oNffhZR8qmpRaojc9
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SBjHfj6DcUOTscfduYPRzMCnL3ibBREIIEU=
-----END CERTIFICATE-----
1 s:C = US, O = "Entrust, Inc.", OU = See www.entrust.net/legal-terms, OU = "(c) 2014 Entrust, Inc. - for authorized use only", CN = Entrust Certification Authority - L1M
i:C = US, O = "Entrust, Inc.", OU = See www.entrust.net/legal-terms, OU = "(c) 2009 Entrust, Inc. - for authorized use only", CN = Entrust Root Certification Authority - G2
-----BEGIN CERTIFICATE-----
MIIFLTCCBBWgAwIBAgIMYaHn0gAAAABR02amMA0GCSqGSIb3DQEBCwUAMIG+MQsw
CQYDVQQGEwJVUzEWMBQGA1UEChMNRW50cnVzdCwgSW5jLjEoMCYGA1UECxMfU2Vl
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-----END CERTIFICATE-----
---
Server certificate
subject=C = ES, ST = Cantabria, L = Santander, jurisdictionC = ES, O = "Grupo Santander (Banco Santander, S.A.)", businessCategory = Private Organization, OU = TOTTA, serialNumber = A39000013, CN = www.santander.pt
issuer=C = US, O = "Entrust, Inc.", OU = See www.entrust.net/legal-terms, OU = "(c) 2014 Entrust, Inc. - for authorized use only", CN = Entrust Certification Authority - L1M
---
No client certificate CA names sent
Peer signing digest: SHA256
Peer signature type: RSA-PSS
Server Temp Key: X25519, 253 bits
---
SSL handshake has read 3802 bytes and written 388 bytes
Verification: OK
---
New, TLSv1.3, Cipher is TLS_AES_256_GCM_SHA384
Server public key is 2048 bit
Secure Renegotiation IS NOT supported
Compression: NONE
Expansion: NONE
No ALPN negotiated
Early data was not sent
Verify return code: 0 (ok)
---
---
Post-Handshake New Session Ticket arrived:
SSL-Session:
Protocol : TLSv1.3
Cipher : TLS_AES_256_GCM_SHA384
Session-ID: 092E17B110E96A42C53877393631A6507B3E39253D6539F7D0A55C9F8DD1AF82
Session-ID-ctx:
Resumption PSK: 8877BAFA943CE3FD7B28B7CA1A8046D360F87100EDB08D899F58037DE44597507B48EB26213A26A2DCE05611DB100B41
PSK identity: None
PSK identity hint: None
SRP username: None
TLS session ticket lifetime hint: 83100 (seconds)
TLS session ticket:
0000 - 00 00 68 4f 95 e5 49 b2-11 15 8b df a1 2a 11 ab ..hO..I......*..
0010 - 56 01 bc b2 6f 4d 45 b9-28 d8 20 a4 8a f5 b5 af V...oME.(. .....
0020 - fd 5d 9b cd ea 62 5d bc-aa af f6 3b b1 87 85 dd .]...b]....;....
0030 - 8b 8c a4 62 dc 53 07 41-16 54 9b 38 b6 58 ca 03 ...b.S.A.T.8.X..
0040 - 00 d9 80 63 cb c8 32 c1-89 41 93 95 6a ec af 5b ...c..2..A..j..[
0050 - 3a b6 c1 f5 91 43 08 43-26 ca 58 33 5f bb b1 1b :....C.C&.X3_...
0060 - af bb 38 65 d4 11 c3 3d-b4 08 56 a4 8b de e7 3b ..8e...=..V....;
0070 - 06 de 96 c2 f1 92 47 98-9e 59 49 f9 b7 17 2b b3 ......G..YI...+.
0080 - ed 21 a1 d0 77 9c 4d 70-7c 97 ca e1 fd f2 57 b7 .!..w.Mp|.....W.
0090 - 77 15 a6 43 77 96 eb 67-8b ec 69 a7 7a 06 72 c8 w..Cw..g..i.z.r.
00a0 - 61 8d ef 07 ba e1 23 5f-c7 b9 c2 64 b5 68 ea c1 a.....#_...d.h..
00b0 - 53 6b 6f 96 cc 07 8d e5-d3 d2 9a 08 fc f4 35 3f Sko...........5?
00c0 - 35 ff 40 b1 3c 69 6d 90-ff db ec 40 54 65 43 19 5.@.<im....@TeC.
00d0 - 79 22 44 9e f3 96 be 0b-a3 bc 2d 22 7f 11 05 16 y"D.......-"....
Start Time: 1639075331
Timeout : 7200 (sec)
Verify return code: 0 (ok)
Extended master secret: no
Max Early Data: 0
---
read R BLOCK
---
Post-Handshake New Session Ticket arrived:
SSL-Session:
Protocol : TLSv1.3
Cipher : TLS_AES_256_GCM_SHA384
Session-ID: 511907D531AF795530321434CFDC7DF2D8FBA18377DAA5E676953A34C3D0348E
Session-ID-ctx:
Resumption PSK: D0008EF7720172350C64F9C398E5D2A331557AB86A50144AE10D5B62E1DD866EBCC0D2F0C88D6EEF192B655B7CCBFBDB
PSK identity: None
PSK identity hint: None
SRP username: None
TLS session ticket lifetime hint: 83100 (seconds)
TLS session ticket:
0000 - 00 00 68 4f 95 e5 49 b2-11 15 8b df a1 2a 11 ab ..hO..I......*..
0010 - ec ae 89 76 9a 0a f8 36-51 41 5c 81 c1 7a 60 b1 ...v...6QA\..z`.
0020 - e6 29 f5 82 8b ee a6 d0-6e fc 64 9d c3 86 4b 5c .)......n.d...K\
0030 - 8d 00 15 6e b2 a7 29 af-27 fd 3e ac e7 51 f1 aa ...n..).'.>..Q..
0040 - af ed 5a 58 15 08 07 de-3d b7 da d5 ce 93 6b 3c ..ZX....=.....k<
0050 - 35 10 7c 79 79 57 21 28-8f c2 f9 61 91 e4 f0 1c 5.|yyW!(...a....
0060 - 33 16 e5 ea e9 b3 66 19-15 05 1a 99 12 5d 8b 5b 3.....f......].[
0070 - 78 21 d4 b6 4e 3a 11 2b-b1 2a 7a ee 39 57 d0 f7 x!..N:.+.*z.9W..
0080 - 5a ad f4 e6 09 1a 52 71-7d 77 a0 03 e8 aa 3a 49 Z.....Rq}w....:I
0090 - 73 4d 8c ed 69 32 87 fa-58 2f eb 4c f5 35 de 50 sM..i2..X/.L.5.P
00a0 - 22 2c bc 1d a6 85 5b 4e-fd a4 e1 57 93 a3 66 67 ",....[N...W..fg
00b0 - 97 2e 47 17 07 c1 de dc-e0 f2 45 4e 77 4d 47 6a ..G.......ENwMGj
00c0 - 2b 16 e8 c7 f1 be b3 b2-16 a2 56 03 68 22 93 bd +.........V.h"..
00d0 - 27 79 1b c2 48 fa 48 7a-3f 21 94 73 62 c6 85 6e 'y..H.Hz?!.sb..n
Start Time: 1639075331
Timeout : 7200 (sec)
Verify return code: 0 (ok)
Extended master secret: no
Max Early Data: 0
---
read R BLOCK
closed
```
With this we created 2 files with the certificate keys given: `c0.pem` and `c1.pem`
### Step 2: Extract the public key (e, n) from the issuer's certificate
We then find the `n` value of the public key using the following command:
```
openssl x509 -in c1.pem -noout -modulus
Modulus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
```
To find the `e` value of the public key, we use the command:
```
openssl x509 -in c1.pem -text -noout > e.txt
```
This give us the following text file:
```
Certificate:
Data:
Version: 3 (0x2)
Serial Number:
61:a1:e7:d2:00:00:00:00:51:d3:66:a6
Signature Algorithm: sha256WithRSAEncryption
Issuer: C = US, O = "Entrust, Inc.", OU = See www.entrust.net/legal-terms, OU = "(c) 2009 Entrust, Inc. - for authorized use only", CN = Entrust Root Certification Authority - G2
Validity
Not Before: Dec 15 15:25:03 2014 GMT
Not After : Oct 15 15:55:03 2030 GMT
Subject: C = US, O = "Entrust, Inc.", OU = See www.entrust.net/legal-terms, OU = "(c) 2014 Entrust, Inc. - for authorized use only", CN = Entrust Certification Authority - L1M
Subject Public Key Info:
Public Key Algorithm: rsaEncryption
RSA Public-Key: (2048 bit)
Modulus:
00:d0:81:c1:39:23:c2:b1:d1:ec:f7:57:dd:55:24:
36:91:20:22:48:f7:fc:ca:52:0a:b0:ab:3f:33:b5:
b0:84:07:f6:df:4e:7a:b0:fb:98:22:3d:01:ac:56:
fb:71:6d:b2:ee:b9:a0:0f:52:77:ab:98:93:be:33:
8a:eb:87:5e:c7:aa:b0:ca:69:8f:43:08:6a:3f:22:
bf:33:39:46:d5:94:f2:e2:4c:05:22:d9:67:80:91:
f1:04:4a:0e:9b:7c:a2:c9:d2:6c:fd:3c:09:84:bd:
fd:6b:14:9a:81:1d:e7:8a:83:ef:61:16:75:47:98:
13:3b:0d:90:16:98:bf:8a:e2:27:32:53:99:99:c3:
fb:96:1c:35:f7:62:ed:8c:bd:49:71:d2:43:43:a1:
a1:e3:21:2a:23:70:a8:75:3d:b2:6c:46:06:61:6f:
18:67:e4:29:7e:b2:3c:c1:c5:5f:09:1e:6e:44:4e:
ec:21:99:58:15:48:f4:55:48:2a:b7:34:b4:05:e3:
7c:49:8c:00:58:de:3a:96:cc:39:dc:61:33:55:ce:
2a:2e:3f:d1:99:62:e8:aa:e6:34:76:31:aa:af:79:
29:96:78:cb:81:14:af:69:da:fb:04:b9:59:83:44:
aa:09:4f:b4:d4:2c:01:9d:9b:94:31:6b:2d:a1:cf:
c1:e5
Exponent: 65537 (0x10001)
X509v3 extensions:
X509v3 Key Usage: critical
Certificate Sign, CRL Sign
X509v3 Extended Key Usage:
TLS Web Client Authentication, TLS Web Server Authentication
X509v3 Basic Constraints: critical
CA:TRUE, pathlen:0
Authority Information Access:
OCSP - URI:http://ocsp.entrust.net
X509v3 CRL Distribution Points:
Full Name:
URI:http://crl.entrust.net/g2ca.crl
X509v3 Certificate Policies:
Policy: X509v3 Any Policy
CPS: http://www.entrust.net/rpa
X509v3 Subject Key Identifier:
C3:F7:D0:B5:2A:30:AD:AF:0D:91:21:70:39:54:DD:BC:89:70:C7:3A
X509v3 Authority Key Identifier:
keyid:6A:72:26:7A:D0:1E:EF:7D:E7:3B:69:51:D4:6C:8D:9F:90:12:66:AB
Signature Algorithm: sha256WithRSAEncryption
b4:87:c7:84:22:1a:29:c0:a4:78:ec:f5:4f:1b:b4:84:97:6f:
77:ee:d4:cf:59:af:a8:43:96:2f:1d:58:de:a6:f3:15:5b:2e:
d9:43:9c:4c:42:b9:17:17:11:ad:b4:b1:41:c7:66:0c:22:86:
05:a7:b4:22:fb:c5:68:68:8a:88:32:d8:71:d6:11:8e:8e:52:
7e:42:09:5d:d9:d9:c0:0e:2c:ef:44:6b:1f:db:24:bc:f5:bc:
b1:aa:c1:8a:e5:66:db:d0:24:f3:df:89:a6:8b:fe:5e:d0:50:
ba:9b:75:00:c9:08:4a:4b:74:3f:58:53:af:f6:b8:a1:6c:84:
98:26:7c:fc:fc:35:51:65:98:1e:3b:6c:76:f0:99:0f:c4:b9:
09:d5:85:d8:24:a7:7f:2a:c2:da:93:4b:64:b8:70:7c:54:29:
d3:82:54:d5:cf:a5:94:c3:33:9e:4d:34:67:e0:38:76:66:00:
8d:d0:2f:c6:ab:09:aa:72:fa:ce:2f:35:89:fc:c0:ed:06:03:
b7:df:8c:25:57:f1:fb:ef:28:88:d9:82:28:c0:04:0f:94:81:
69:23:e0:91:28:c5:5e:d5:0c:d7:d4:20:3b:ef:c9:f8:c0:8f:
a7:41:cd:57:44:6e:6c:67:62:fe:84:df:e7:60:d8:ef:bf:d4:
22:1e:0d:e7
```
In this file we can find the `Exponent` used in the certificate:
```
Exponent: 65537 (0x10001)
```
### Step 3: Extract the signature from the server's certificate
For this we use the following command:
```bash
openssl x509 -in c0.pem -text -noout
```
Obtaining the following signature:
```
Signature Algorithm: sha256WithRSAEncryption
66:b4:1e:fa:35:f6:44:86:2c:2a:1c:77:fb:1b:22:d4:2a:e6:
70:9b:39:64:74:36:ca:12:fb:a5:e0:b5:fa:b7:b9:fd:5a:73:
a6:a4:98:ed:ed:4d:71:24:22:ec:ba:db:a2:9b:6a:0b:27:a5:
54:ce:ea:4a:e1:86:54:16:34:a5:45:75:19:c2:58:89:7b:07:
4d:a8:35:f7:e1:65:1f:2a:9a:94:5a:a2:37:3d:3c:dd:cb:e5:
45:b1:be:d1:94:67:e8:cf:a6:04:1d:45:38:05:ce:5d:71:97:
92:c9:06:35:54:8e:3f:a1:95:1f:0d:a4:da:a4:bf:04:d6:c8:
d9:f7:84:97:a8:4a:d5:28:1f:9f:ca:b7:26:48:99:42:b0:b3:
2a:e6:e0:38:bf:6c:c8:8e:ae:e7:df:4c:a3:e0:f5:5e:e1:41:
31:db:bf:62:fa:72:41:2c:86:92:49:95:bf:2d:d0:23:ea:ef:
b3:53:56:62:c9:9d:b9:b9:86:24:f7:63:97:a8:92:10:83:66:
b8:81:4a:b8:ec:ac:37:d6:c6:d0:81:71:17:46:27:7f:34:e3:
39:f5:3d:3c:f1:ea:08:87:2c:0e:f2:a7:a5:08:48:18:c7:7e:
3e:83:71:43:93:b1:c7:dd:b9:83:d1:cc:c0:a7:2f:78:9b:05:
11:08:20:45
```
Using the command `cat signature | tr -d '[:space:]:'` on the signature value we obtain the following key:
```
66b41efa35f644862c2a1c77fb1b22d42ae6
709b39647436ca12fba5e0b5fab7b9fd5a73
a6a498eded4d712422ecbadba29b6a0b27a5
54ceea4ae186541634a5457519c258897b07
4da835f7e1651f2a9a945aa2373d3cddcbe5
45b1bed19467e8cfa6041d453805ce5d7197
92c90635548e3fa1951f0da4daa4bf04d6c8
d9f78497a84ad5281f9fcab726489942b0b3
2ae6e038bf6cc88eaee7df4ca3e0f55ee141
31dbbf62fa72412c86924995bf2dd023eaef
b3535662c99db9b98624f76397a892108366
b8814ab8ecac37d6c6d081711746277f34e3
39f53d3cf1ea08872c0ef2a7a5084818c77e
3e83714393b1c7ddb983d1ccc0a72f789b05
11082045
```
### Step 4: Extract the body of the server's certificate
In order to view the information in the server's certificate we use the following command:
```bash
openssl asn1parse -i -in c0.pem
```
Executing the command we obtain this text:
```
0:d=0 hl=4 l=1894 cons: SEQUENCE
4:d=1 hl=4 l=1614 cons: SEQUENCE
8:d=2 hl=2 l= 3 cons: cont [ 0 ]
10:d=3 hl=2 l= 1 prim: INTEGER :02
13:d=2 hl=2 l= 16 prim: INTEGER :50F1F0BE71FDF0F9E39208C7BE5F9746
31:d=2 hl=2 l= 13 cons: SEQUENCE
33:d=3 hl=2 l= 9 prim: OBJECT :sha256WithRSAEncryption
44:d=3 hl=2 l= 0 prim: NULL
46:d=2 hl=3 l= 186 cons: SEQUENCE
49:d=3 hl=2 l= 11 cons: SET
51:d=4 hl=2 l= 9 cons: SEQUENCE
53:d=5 hl=2 l= 3 prim: OBJECT :countryName
58:d=5 hl=2 l= 2 prim: PRINTABLESTRING :US
62:d=3 hl=2 l= 22 cons: SET
64:d=4 hl=2 l= 20 cons: SEQUENCE
66:d=5 hl=2 l= 3 prim: OBJECT :organizationName
71:d=5 hl=2 l= 13 prim: PRINTABLESTRING :Entrust, Inc.
86:d=3 hl=2 l= 40 cons: SET
88:d=4 hl=2 l= 38 cons: SEQUENCE
90:d=5 hl=2 l= 3 prim: OBJECT :organizationalUnitName
95:d=5 hl=2 l= 31 prim: PRINTABLESTRING :See www.entrust.net/legal-terms
128:d=3 hl=2 l= 57 cons: SET
130:d=4 hl=2 l= 55 cons: SEQUENCE
132:d=5 hl=2 l= 3 prim: OBJECT :organizationalUnitName
137:d=5 hl=2 l= 48 prim: PRINTABLESTRING :(c) 2014 Entrust, Inc. - for authorized use only
187:d=3 hl=2 l= 46 cons: SET
189:d=4 hl=2 l= 44 cons: SEQUENCE
191:d=5 hl=2 l= 3 prim: OBJECT :commonName
196:d=5 hl=2 l= 37 prim: PRINTABLESTRING :Entrust Certification Authority - L1M
235:d=2 hl=2 l= 30 cons: SEQUENCE
237:d=3 hl=2 l= 13 prim: UTCTIME :210624103419Z
252:d=3 hl=2 l= 13 prim: UTCTIME :220701103419Z
267:d=2 hl=3 l= 218 cons: SEQUENCE
270:d=3 hl=2 l= 11 cons: SET
272:d=4 hl=2 l= 9 cons: SEQUENCE
274:d=5 hl=2 l= 3 prim: OBJECT :countryName
279:d=5 hl=2 l= 2 prim: PRINTABLESTRING :ES
283:d=3 hl=2 l= 18 cons: SET
285:d=4 hl=2 l= 16 cons: SEQUENCE
287:d=5 hl=2 l= 3 prim: OBJECT :stateOrProvinceName
292:d=5 hl=2 l= 9 prim: PRINTABLESTRING :Cantabria
303:d=3 hl=2 l= 18 cons: SET
305:d=4 hl=2 l= 16 cons: SEQUENCE
307:d=5 hl=2 l= 3 prim: OBJECT :localityName
312:d=5 hl=2 l= 9 prim: PRINTABLESTRING :Santander
323:d=3 hl=2 l= 19 cons: SET
325:d=4 hl=2 l= 17 cons: SEQUENCE
327:d=5 hl=2 l= 11 prim: OBJECT :jurisdictionCountryName
340:d=5 hl=2 l= 2 prim: PRINTABLESTRING :ES
344:d=3 hl=2 l= 48 cons: SET
346:d=4 hl=2 l= 46 cons: SEQUENCE
348:d=5 hl=2 l= 3 prim: OBJECT :organizationName
353:d=5 hl=2 l= 39 prim: PRINTABLESTRING :Grupo Santander (Banco Santander, S.A.)
394:d=3 hl=2 l= 29 cons: SET
396:d=4 hl=2 l= 27 cons: SEQUENCE
398:d=5 hl=2 l= 3 prim: OBJECT :businessCategory
403:d=5 hl=2 l= 20 prim: PRINTABLESTRING :Private Organization
425:d=3 hl=2 l= 14 cons: SET
427:d=4 hl=2 l= 12 cons: SEQUENCE
429:d=5 hl=2 l= 3 prim: OBJECT :organizationalUnitName
434:d=5 hl=2 l= 5 prim: PRINTABLESTRING :TOTTA
441:d=3 hl=2 l= 18 cons: SET
443:d=4 hl=2 l= 16 cons: SEQUENCE
445:d=5 hl=2 l= 3 prim: OBJECT :serialNumber
450:d=5 hl=2 l= 9 prim: PRINTABLESTRING :A39000013
461:d=3 hl=2 l= 25 cons: SET
463:d=4 hl=2 l= 23 cons: SEQUENCE
465:d=5 hl=2 l= 3 prim: OBJECT :commonName
470:d=5 hl=2 l= 16 prim: PRINTABLESTRING :www.santander.pt
488:d=2 hl=4 l= 290 cons: SEQUENCE
492:d=3 hl=2 l= 13 cons: SEQUENCE
494:d=4 hl=2 l= 9 prim: OBJECT :rsaEncryption
505:d=4 hl=2 l= 0 prim: NULL
507:d=3 hl=4 l= 271 prim: BIT STRING
782:d=2 hl=4 l= 836 cons: cont [ 3 ]
786:d=3 hl=4 l= 832 cons: SEQUENCE
790:d=4 hl=2 l= 12 cons: SEQUENCE
792:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Basic Constraints
797:d=5 hl=2 l= 1 prim: BOOLEAN :255
800:d=5 hl=2 l= 2 prim: OCTET STRING [HEX DUMP]:3000
804:d=4 hl=2 l= 29 cons: SEQUENCE
806:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Subject Key Identifier
811:d=5 hl=2 l= 22 prim: OCTET STRING [HEX DUMP]:0414981F000A44C8B5AB97266D82A880E28A824BFAE9
835:d=4 hl=2 l= 31 cons: SEQUENCE
837:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Authority Key Identifier
842:d=5 hl=2 l= 24 prim: OCTET STRING [HEX DUMP]:30168014C3F7D0B52A30ADAF0D9121703954DDBC8970C73A
868:d=4 hl=2 l= 104 cons: SEQUENCE
870:d=5 hl=2 l= 8 prim: OBJECT :Authority Information Access
880:d=5 hl=2 l= 92 prim: OCTET STRING [HEX DUMP]:305A302306082B060105050730018617687474703A2F2F6F6373702E656E74727573742E6E6574303306082B060105050730028627687474703A2F2F6169612E656E74727573742E6E65742F6C316D2D636861696E3235362E636572
974:d=4 hl=2 l= 51 cons: SEQUENCE
976:d=5 hl=2 l= 3 prim: OBJECT :X509v3 CRL Distribution Points
981:d=5 hl=2 l= 44 prim: OCTET STRING [HEX DUMP]:302A3028A026A0248622687474703A2F2F63726C2E656E74727573742E6E65742F6C6576656C316D2E63726C
1027:d=4 hl=2 l= 83 cons: SEQUENCE
1029:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Subject Alternative Name
1034:d=5 hl=2 l= 76 prim: OCTET STRING [HEX DUMP]:304A82107777772E73616E74616E6465722E7074820C73616E74616E6465722E7074821173616E74616E646572746F7474612E707482157777772E73616E74616E646572746F7474612E7074
1112:d=4 hl=2 l= 14 cons: SEQUENCE
1114:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Key Usage
1119:d=5 hl=2 l= 1 prim: BOOLEAN :255
1122:d=5 hl=2 l= 4 prim: OCTET STRING [HEX DUMP]:030205A0
1128:d=4 hl=2 l= 29 cons: SEQUENCE
1130:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Extended Key Usage
1135:d=5 hl=2 l= 22 prim: OCTET STRING [HEX DUMP]:301406082B0601050507030106082B06010505070302
1159:d=4 hl=2 l= 75 cons: SEQUENCE
1161:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Certificate Policies
1166:d=5 hl=2 l= 68 prim: OCTET STRING [HEX DUMP]:30423037060A6086480186FA6C0A01023029302706082B06010505070201161B68747470733A2F2F7777772E656E74727573742E6E65742F7270613007060567810C0101
1236:d=4 hl=4 l= 382 cons: SEQUENCE
1240:d=5 hl=2 l= 10 prim: OBJECT :CT Precertificate SCTs
1252:d=5 hl=4 l= 366 prim: OCTET STRING [HEX DUMP]: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
1622:d=1 hl=2 l= 13 cons: SEQUENCE
1624:d=2 hl=2 l= 9 prim: OBJECT :sha256WithRSAEncryption
1635:d=2 hl=2 l= 0 prim: NULL
1637:d=1 hl=4 l= 257 prim: BIT STRING
```
From this text we can extract the body of the certificate using the command:
```bash
openssl anslparse -i -in c0.pem -strparse 4 -out c0_body.bin -nout
```
Running the command we obtain a bin file `c0_body.bin` with the body of the certificate.
After this we calculate the hash value of the body using the `sha256sum` algorithm:
```bash
sha256sum c0_body.bin
```
Output:
```
793ec98a8800cb7b6ab2354a15b1b911972ac036d2607936e00aae2ffe1a5e68 c0_body.bin
```
### Step 5: Verify the signature
To calculate the body of the server using the signature and the values of e and n, we just need to change the initial valeus of these variables in the code presented in task 5:
```c=
BN_hex2bn(&n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
BN_hex2bn(&e, "010001");
BN_hex2bn(&M, "793ec98a8800cb7b6ab2354a15b1b911972ac036d2607936e00aae2ffe1a5e68 ");
BN_hex2bn(&S, "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");
BN_mod_exp(c, S, e, n, ctx);
printBN("M given = ", M);
printBN("M obtained =", c);
```
This gave us the following result:
```
M given = 793EC98A8800CB7B6AB2354A15B1B911972AC036D2607936E00AAE2FFE1A5E68
M obtained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
```
At first it might seem that the value found is different than the body of the server's certificate, but with further inspection we can see that its same value only padded to meet the same number of bytes as the signature and n.
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