Private-Key: (2048 bit) modulus: 00:c2:5e:ce:82:de:13:96:65:95:4b:87:a2:3a:54: c

Question

Private-Key: (2048 bit)

modulus:

00:c2:5e:ce:82:de:13:96:65:95:4b:87:a2:3a:54:

c8:b9:e5:00:96:c6:0b:a5:c7:5c:e4:54:35:cc:43:

4d:a8:cd:cb:67:52:52:e3:c2:55:25:ae:48:c9:7f:

69:e6:46:7d:d5:08:2d:6f:ec:e5:7f:98:7e:fa:04:

df:f5:28:f3:7b:89:5e:5a:38:e0:30:f6:66:85:72:

ed:2e:ca:a7:c1:20:ac:ee:75:02:5b:43:52:32:2e:

96:b6:1e:9e:02:f0:d3:0b:20:f0:87:12:10:4d:06:

58:d0:ca:8c:f3:62:39:06:ff:9c:96:20:2c:b3:07:

2c:c5:7d:d3:5b:22:e3:ce:5c:8b:d7:1d:4c:53:e4:

2d:52:18:4d:7a:ca:9c:1f:7f:99:be:19:07:94:37:

62:8c:37:64:6a:47:60:62:20:6c:84:ff:e5:28:2e:

ad:64:7d:e5:7b:a2:65:a8:04:d4:ba:48:c3:6c:32:

9c:0b:c9:b1:e6:7d:13:9e:3d:63:0c:b3:bc:4c:ee:

05:1b:a6:b9:37:1f:d6:7a:9d:8e:28:3b:cb:35:eb:

8c:a3:a1:bf:71:72:fd:58:12:a4:51:91:2d:c4:db:

8c:6e:4e:68:4d:0c:59:0d:c5:0b:7c:9f:20:a3:ef:

fd:2c:15:7b:f3:4f:61:f6:f2:78:5e:c2:7d:31:ee:

8d:59

publicExponent: 65537 (0x10001)

privateExponent:

14:7c:1f:71:a5:6c:15:0e:b1:b4:5b:20:1a:db:b2:

a7:fc:dd:9a:06:57:55:ea:73:fc:d0:6f:37:0e:e4:

d1:64:97:0a:55:9f:cc:c7:3f:68:10:3d:35:97:4b:

16:ec:d8:f8:29:bb:78:2f:54:64:a5:32:c4:b0:dc:

d5:9e:b2:42:93:d0:b0:db:8b:ad:3d:5a:c2:b7:c1:

0d:8c:68:2f:af:7c:ea:48:c4:ad:cf:a3:8b:0c:63:

ef:df:bc:31:7d:26:10:e7:30:d2:5b:72:02:5c:27:

e2:21:66:c7:83:d3:94:cc:3d:b9:88:bc:8d:45:b5:

4f:3f:1a:96:61:4f:f7:00:94:e2:6b:89:32:9d:37:

24:6e:4d:49:7a:73:02:70:ac:f1:3b:99:51:d6:4b:

e1:42:7a:fc:e3:d1:2b:b1:74:a8:97:5b:00:da:6b:

f0:e6:d4:86:2d:f5:41:cb:4c:e3:eb:fa:53:3e:29:

f3:dd:d3:fa:90:e1:d3:4b:42:d4:9c:48:05:48:53:

2d:12:d0:c3:07:a4:f5:27:4f:df:48:e0:d1:a2:c2:

c2:03:72:0a:ff:6a:26:38:b0:f7:41:3f:bb:0c:72:

18:e9:6d:27:34:e2:39:15:b4:19:7e:c6:96:49:72:

70:6c:79:02:98:1a:5e:5a:22:99:46:47:90:d4:fd:

8d

prime1:

00:e7:e4:27:51:2e:cc:d8:79:23:8e:c8:74:91:9f:

6c:3c:38:f3:f8:be:bd:fc:0c:d3:29:cc:9f:ac:e5:

3b:b2:7c:98:5e:a5:38:0a:3b:4d:a9:cc:fe:ed:ff:

84:a7:b1:0e:5e:b6:48:fc:0f:50:e0:24:8a:7b:2d:

5a:54:eb:2b:c9:c6:fa:5a:f2:d3:38:2d:40:fa:da:

e4:e0:a7:61:f8:d1:40:be:a7:47:51:27:e7:c6:c7:

5e:03:71:44:9e:79:0b:9e:69:af:da:3d:7b:d8:38:

07:0b:fd:ce:28:f3:98:47:9f:3d:1d:5f:d0:8b:39:

21:dc:45:10:1c:14:7b:fe:93

prime2:

00:d6:94:06:57:21:bd:75:eb:ea:90:14:15:4f:38:

84:5b:db:09:e8:1f:81:44:b6:3b:8f:8d:6d:a8:2a:

1b:4f:07:1b:4c:d5:1e:44:27:7d:f0:b3:9f:d1:a8:

8d:dd:3a:b6:95:cf:e0:50:8e:a8:ef:18:fa:8c:b8:

e6:16:fe:76:e9:62:78:0a:2d:59:29:92:d7:ed:6c:

0b:5d:c1:c2:48:4f:b4:63:bb:68:eb:b1:e6:0d:1a:

b3:40:07:c3:9b:ff:7d:c0:f8:e3:44:d6:13:ba:fd:

41:45:76:ed:22:0c:ed:45:9f:44:e9:d2:92:ed:c7:

3f:5b:be:0a:db:4f:76:8b:e3

exponent1:

00:e0:d5:f7:45:20:82:5f:35:7d:43:78:9a:85:83:

34:97:63:85:31:de:07:28:f6:6a:5b:71:da:33:f2:

03:22:b8:e9:a7:82:bc:46:37:5f:c2:ce:4f:33:05:

77:62:7c:2a:79:ac:76:31:66:cc:3c:b7:51:49:c3:

1c:7d:8f:93:62:4d:b9:fa:2a:cc:9f:00:49:ab:b0:

47:10:87:5a:57:bc:d3:f0:c7:12:3f:de:d8:3c:84:

9b:63:ed:13:e2:2f:6d:13:88:57:87:db:eb:53:26:

67:b6:b9:bc:85:3a:42:a0:d7:05:93:90:8e:0c:1f:

35:f4:73:45:fe:63:e2:21:5f

exponent2:

00:c6:c5:46:7e:4e:55:f8:37:7b:08:ce:61:d3:09:

f2:aa:38:64:9b:ef:1b:4b:30:58:4a:29:63:cc:24:

7d:39:9e:c6:ae:4c:b9:85:be:e3:8e:26:86:c4:ab:

94:06:bc:4d:c8:c7:50:ee:ae:c8:54:fc:59:94:23:

ef:98:9b:4e:41:f1:a2:6b:23:08:f6:56:fd:c9:04:

8e:1e:e2:ce:fb:37:42:4e:24:9c:13:c1:ae:ee:b2:

ae:1b:81:55:b5:5b:5d:fd:75:50:ea:42:38:f2:bd:

89:4a:34:d9:83:f2:5f:95:5f:ed:71:80:4b:f5:11:

de:62:38:d0:4b:c9:f7:71:ab

coefficient:

1f:8e:f6:65:3e:ce:09:4d:2f:e2:2e:a4:0d:37:fc:

11:85:8c:35:c2:1d:5b:32:21:de:08:8c:87:61:11:

e7:7f:98:f6:bc:84:ae:f9:b4:31:14:fd:8e:ab:32:

f9:95:8c:a6:7e:b9:57:72:3f:48:ee:ee:5a:3d:b5:

9d:3f:0a:05:1b:b9:1b:bc:ae:0c:a3:1e:1f:2a:fe:

e8:1f:e6:65:64:68:6f:71:99:7c:44:49:3d:46:8b:

9a:58:d7:04:a5:3d:9d:c7:11:56:34:fd:1d:97:cc:

1a:f2:48:09:28:75:12:8c:33:c3:36:71:92:ff:df:

ca:c8:3e:2d:20:61:26:d3

writing RSA key

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

MIIEpAIBAAKCAQEAwl7Ogt4TlmWVS4eiOlTIueUAlsYLpcdc5FQ1zENNqM3LZ1JS

48JVJa5IyX9p5kZ91Qgtb+zlf5h++gTf9Sjze4leWjjgMPZmhXLtLsqnwSCs7nUC

W0NSMi6Wth6eAvDTCyDwhxIQTQZY0MqM82I5Bv+cliAsswcsxX3TWyLjzlyL1x1M

U+QtUhhNesqcH3+ZvhkHlDdijDdkakdgYiBshP/lKC6tZH3le6JlqATUukjDbDKc

C8mx5n0Tnj1jDLO8TO4FG6a5Nx/Wep2OKDvLNeuMo6G/cXL9WBKkUZEtxNuMbk5o

TQxZDcULfJ8go+/9LBV7809h9vJ4XsJ9Me6NWQIDAQABAoIBABR8H3GlbBUOsbRb

IBrbsqf83ZoGV1Xqc/zQbzcO5NFklwpVn8zHP2gQPTWXSxbs2Pgpu3gvVGSlMsSw

3NWeskKT0LDbi609WsK3wQ2MaC+vfOpIxK3Po4sMY+/fvDF9JhDnMNJbcgJcJ+Ih

ZseD05TMPbmIvI1FtU8/GpZhT/cAlOJriTKdNyRuTUl6cwJwrPE7mVHWS+FCevzj

0SuxdKiXWwDaa/Dm1IYt9UHLTOPr+lM+KfPd0/qQ4dNLQtScSAVIUy0S0MMHpPUn

T99I4NGiwsIDcgr/aiY4sPdBP7sMchjpbSc04jkVtBl+xpZJcnBseQKYGl5aIplG

R5DU/Y0CgYEA5+QnUS7M2Hkjjsh0kZ9sPDjz+L69/AzTKcyfrOU7snyYXqU4CjtN

qcz+7f+Ep7EOXrZI/A9Q4CSKey1aVOsrycb6WvLTOC1A+trk4Kdh+NFAvqdHUSfn

xsdeA3FEnnkLnmmv2j172DgHC/3OKPOYR589HV/Qizkh3EUQHBR7/pMCgYEA1pQG

VyG9devqkBQVTziEW9sJ6B+BRLY7j41tqCobTwcbTNUeRCd98LOf0aiN3Tq2lc/g

UI6o7xj6jLjmFv526WJ4Ci1ZKZLX7WwLXcHCSE+0Y7to67HmDRqzQAfDm/99wPjj

RNYTuv1BRXbtIgztRZ9E6dKS7cc/W74K2092i+MCgYEA4NX3RSCCXzV9Q3iahYM0

l2OFMd4HKPZqW3HaM/IDIrjpp4K8Rjdfws5PMwV3Ynwqeax2MWbMPLdRScMcfY+T

Yk25+irMnwBJq7BHEIdaV7zT8McSP97YPISbY+0T4i9tE4hXh9vrUyZntrm8hTpC

oNcFk5CODB819HNF/mPiIV8CgYEAxsVGfk5V+Dd7CM5h0wnyqjhkm+8bSzBYSilj

zCR9OZ7Grky5hb7jjiaGxKuUBrxNyMdQ7q7IVPxZlCPvmJtOQfGiayMI9lb9yQSO

HuLO+zdCTiScE8Gu7rKuG4FVtVtd/XVQ6kI48r2JSjTZg/JflV/tcYBL9RHeYjjQ

S8n3casCgYAfjvZlPs4JTS/iLqQNN/wRhYw1wh1bMiHeCIyHYRHnf5j2vISu+bQx

FP2OqzL5lYymfrlXcj9I7u5aPbWdPwoFG7kbvK4Mox4fKv7oH+ZlZGhvcZl8REk9

RouaWNcEpT2dxxFWNP0dl8wa8kgJKHUSjDPDNnGS/9/KyD4tIGEm0w==

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

The private key file contains the following data fields:

1. modulus
2. public exponent
3. private exponent
4. prime1
5. prime2
6. exponent1
7. exponent2
8. coefficient

explain what all of the above fields are for

Explanation / Answer

Construction of RSA private key starts by choosing prime1 and prime2 numbers.

prime1) prime1 is a randomly chosen prime number (for better security it should not be too small).

prime2) prime2 is also a randomly chosen prime number (for better security it should not be too small and prime1 and prime2 should be similar in magnitude and should differ in length by few digits).

modulus) Modulus is multiplication of prime1 and prime2 ( modulus = prime1*prime2). It is used as modulus in public key and private key.Its length expressed in bits is denoted as the key length of the private key.

public exponent) public exponent e is chosen such that 1 < e < ?(n) and gcd(e, ?(n)) = 1; i.e., e and ?(n) are coprime. where ? is totient function.

private exponent) private exponent d is the modular multiplicative inverse of e (modulo ?(n)).So d ? e?1 (mod ?(n)).

From above-mentioned fields, both encryption and decryption operation can be carried out but to improve the performance Chinese remainder theorem is used. To use Chinese remainder theorem we need exponent1, exponent2, and coefficient.

exponent1) exponent1 dp is used in Chinese remainder theorem for fast computation. Here dp = d (mod p?1 ).

exponent2) exponent2 dq is used in Chinese remainder theorem. Here dq = d ( mod q - 1).

coefficient) Coefficient used in Chinese remainder theorem.  qinv = q-1 ( mod p).

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