ssh の RSA キーファイルから素数を取り出してみた

RSA 暗号は大きな数の素因数分解が困難であることを利用したアルゴリズムです。このため、秘密鍵には2つの大きな素数が含まれています。以前、これを確かめるためにわざと鍵長を短く (32bit) にして、その動作を観察したことがありましたが、ふと思い立って普段使っている鍵にどれくらい大きな素数が使われているのか調べてみました。僕自身、自分の鍵の素数を確認してみるというのは初めての経験です。 † openssl の標準のコマンドでは10進表示はできないあらかじめ断っておきますが、このエントリで使われている秘密鍵はサンプルとして生成したものです。自分が実際に使っている鍵に含まれているこれらの値を公開してはいけません。 OpenSSL では openssl rsa -text -noout を使うと、秘密鍵の情報を表示させることができるのですが、鍵長が一定以上になると値が16進数で表示されるので、素数の値を直感的に理解することが難しくなります。具体的には以下のような感じになります。このうち、秘密鍵にあたる素数は prime1, prime2 です︵ただし、実際の計算に使われる秘密鍵にあたる値は publicExponent^(-1) mod ( prime1 - 1 ) * ( prime2 - 1 ) を計算して得られる privateExponent の方です︶。

$ ssh-keygen ssh-keygen -t rsa -b 4096 -f ./id_rsa
$ openssl rsa -text -noout <  id_rsa
Private-Key: (4096 bit)
modulus:
    00:d9:75:60:e5:03:be:f0:42:7b:b0:7e:8e:dc:76:
    92:8c:da:6c:85:64:e6:42:67:a2:c5:b7:4f:7e:f3:
    1e:4e:52:cd:1f:6d:81:5a:f6:a6:1e:a5:9e:b5:37:
    af:6b:98:5f:13:1f:cf:ae:f0:69:9b:a2:15:ce:bc:
    f9:55:62:b8:d2:c2:4a:f7:6a:96:14:3f:10:da:b6:
    38:2a:a9:95:19:d3:b8:42:56:a8:19:bc:fc:80:25:
    70:25:2a:67:08:ba:c3:26:eb:ef:a9:d4:cd:99:3c:
    51:ba:56:a6:b9:a2:e0:2f:f5:af:5c:62:2d:83:97:
    cf:fe:d0:36:55:52:fc:f7:ff:5d:19:d7:f5:c2:c1:
    93:06:52:fb:4b:a4:21:fb:25:f2:b1:9f:df:09:66:
    8f:bc:4a:3a:f1:97:6f:fa:ca:a2:67:95:4f:d4:a8:
    c4:84:6a:af:8e:ee:1d:01:7d:1c:2e:69:ed:c8:ac:
    d4:d7:8d:d4:95:86:51:42:6b:b3:d4:0e:86:3a:62:
    bd:d4:cf:a0:c0:03:4d:35:e8:3a:c3:f6:2e:fa:29:
    b5:ea:46:83:5a:06:b5:36:84:b6:90:4a:06:2a:30:
    1f:aa:a9:e2:ae:e8:27:25:c8:49:2b:3f:73:bd:19:
    8f:c7:52:32:47:91:61:cf:2b:ca:cc:39:56:32:5b:
    a8:2e:96:d6:4f:dc:cb:97:0e:0f:e7:0e:a2:d1:19:
    9f:c0:91:9f:e0:19:eb:a5:57:c3:58:ac:dd:d1:05:
    6e:ce:61:52:20:d7:7f:62:09:07:5f:4e:95:f3:51:
    04:c2:f3:cc:5e:60:7b:8a:a6:c3:51:87:9a:1f:da:
    0a:f9:81:22:e3:00:c8:22:3e:53:ea:17:35:77:63:
    fb:34:3f:a8:5f:75:ae:d5:1b:3f:8f:7f:89:3f:4d:
    f4:40:15:56:9b:7b:db:95:73:77:85:80:7e:ec:f8:
    57:e7:08:d2:c9:51:54:40:b0:04:e9:ee:7c:b4:6d:
    06:61:31:3a:78:bd:2d:b9:98:b1:ea:5a:ec:4d:fb:
    99:3c:28:0e:c7:c9:e4:c9:31:9b:8d:43:45:88:f8:
    92:da:c1:70:84:f7:54:9b:99:07:80:3d:f4:69:82:
    eb:0b:d5:a6:74:ff:bf:d8:82:24:b5:07:b5:ac:43:
    db:10:04:fd:99:93:58:ba:28:d6:57:ec:ad:69:07:
    87:d9:c9:67:d6:fe:32:d5:3b:65:b1:59:75:36:2e:
    a2:d0:8d:91:2c:75:4b:f0:67:69:d4:68:d0:70:91:
    da:f7:d0:c0:f1:e8:5a:ac:09:63:4c:7b:b0:30:fa:
    32:4b:f9:ff:15:f2:a2:c7:4b:e0:1a:a1:0b:fd:ea:
    ae:dd:11
publicExponent: 65537 (0x10001)
privateExponent:
    24:25:bc:db:94:c9:d2:8a:5a:a1:36:08:6d:61:50:
    6f:15:87:e0:8c:ab:31:f2:a4:3e:15:80:a3:6b:aa:
    67:fe:9f:96:4f:cf:a6:dd:45:33:e6:f8:00:cb:a6:
    df:4d:ce:d0:11:03:a6:e2:d3:cb:49:20:32:e3:7a:
    d6:d4:b8:40:60:87:b1:a5:08:19:89:7a:35:c1:00:
    89:26:f3:5f:fd:af:ad:67:56:3e:21:0a:20:0b:19:
    1f:34:73:38:22:43:7a:5e:d1:d1:fb:4b:c9:e3:1a:
    fa:cc:98:9e:57:12:c9:26:1f:e6:4d:73:9f:62:46:
    f2:e6:8d:81:95:6f:f3:91:81:5a:60:69:bb:dc:0d:
    a2:89:7c:b8:50:2a:4d:e3:3a:fa:87:c6:bd:7b:4c:
    fd:51:a1:86:b3:53:ab:ee:af:1c:64:d7:86:54:1f:
    86:9b:6d:5f:66:f9:6f:5b:ea:a9:54:76:70:5b:3a:
    6e:32:67:5a:0b:9f:ca:58:b8:e5:d1:1c:42:ec:52:
    fd:e3:1d:a7:8d:64:f2:47:0e:2d:b0:e8:92:31:08:
    ca:4e:ba:5a:e6:ae:8c:c7:76:c9:79:64:98:f2:b2:
    f0:dd:98:8e:d6:8c:5b:5d:75:bc:38:a9:f6:8f:1f:
    35:54:d6:82:cb:09:e5:34:34:5b:ca:54:27:be:d0:
    af:cf:4e:36:be:e5:51:27:9b:a5:95:e2:eb:ce:06:
    2d:f9:2d:15:59:bb:bd:72:c0:a7:1c:8b:fd:6f:ca:
    9f:fa:e4:1d:cb:a2:f4:3e:c5:96:51:d7:bf:48:bc:
    31:66:cf:90:ad:73:49:7e:60:c8:eb:9e:4b:3b:52:
    81:e7:f3:20:8d:9f:6c:fe:82:20:6e:0c:cc:5e:8a:
    90:15:cf:a9:49:50:d2:86:97:19:7b:bb:38:20:39:
    89:9a:a3:58:a7:1c:79:01:88:d6:b7:97:ee:b6:87:
    0b:57:fe:4d:f2:2c:39:f8:84:a0:f3:75:c9:43:ed:
    46:91:15:70:c1:a3:41:74:b3:9f:ab:16:64:a7:bd:
    cd:fe:42:53:34:d3:20:49:cc:6a:f5:94:a1:6a:1d:
    ce:98:c7:3c:ef:dd:90:ec:33:42:b0:62:9f:6b:7c:
    a7:6a:31:14:f2:1e:eb:8f:ec:27:30:af:ee:bb:7a:
    c6:c8:75:01:10:4b:d5:67:e6:15:5f:49:12:7e:15:
    15:a4:74:ea:c0:ab:46:3f:0c:55:07:8e:c4:9e:f5:
    1e:e9:8c:93:39:e8:10:54:e8:9c:3b:cc:a9:40:63:
    b6:5a:a3:99:a4:22:5e:87:57:3c:34:f9:3c:20:f3:
    da:c5:c4:1d:62:28:5c:67:99:9f:f9:37:d6:a1:50:
    8e:21
prime1:
    00:fb:45:a9:e2:f1:31:90:62:f7:ca:b8:8f:cb:4f:
    94:73:21:7b:2b:a9:15:8c:a1:a6:4e:07:ad:45:17:
    d6:04:91:cd:c0:56:6f:ac:22:4c:ee:06:29:e0:0b:
    4a:51:9d:1a:8f:ca:42:37:f1:e9:77:e3:9d:36:7d:
    8b:f5:4c:f8:28:89:16:a3:7c:21:9c:d5:6f:ad:f3:
    83:91:6f:f1:c9:66:8b:40:76:5e:78:1f:15:b6:ee:
    1d:2c:20:e1:dd:de:9d:cf:ae:73:a6:46:51:bf:dc:
    25:35:3c:3f:c7:03:9f:1e:7e:8b:a8:17:47:86:71:
    5c:ea:7b:a1:15:10:67:9d:83:46:3e:89:68:dc:7d:
    b4:0d:2c:89:f6:c3:b0:ea:69:72:7f:a3:30:88:b1:
    e2:ef:4b:04:cc:d1:e9:07:a1:22:b4:d9:52:1f:24:
    83:c9:5b:0c:86:c1:c7:17:fe:08:64:5e:47:cf:86:
    a3:d1:37:7a:98:67:62:04:47:ec:fa:1a:9e:ef:cc:
    2f:5e:25:f7:36:c4:17:55:dc:32:21:b4:98:d2:ff:
    8b:92:50:e9:4d:d8:46:56:63:84:5d:19:8e:31:31:
    45:02:14:84:33:f8:e8:0d:39:f6:49:88:69:10:a9:
    85:d7:ec:2b:b2:3a:85:17:7d:56:d3:a6:08:46:c2:
    24:45
prime2:
    00:dd:8c:d7:1c:35:1a:21:7e:30:b2:ad:a2:6d:e5:
    7c:74:55:ba:72:d1:14:75:4f:6d:e9:a4:ca:b1:d0:
    d7:9d:a7:5a:31:6a:16:d7:0c:26:15:38:c4:36:91:
    5d:de:2d:c5:e1:fd:1d:b8:20:fd:bd:dc:3a:e1:a4:
    b5:d1:a6:a0:0d:7e:f2:d6:99:34:ce:18:e9:69:a2:
    c5:1d:b4:da:af:a9:e8:64:d4:2f:f9:b8:bf:3e:e8:
    f4:54:dc:59:d2:e1:c0:a6:cc:06:53:e2:73:7f:08:
    9d:3c:c7:52:ae:97:f8:a3:8e:e3:2f:29:64:36:77:
    71:71:ef:d1:85:86:01:c1:89:6e:e4:5c:e2:a0:9f:
    ca:8f:26:72:db:dc:f7:d3:d8:5d:ec:c4:71:60:27:
    12:a4:a7:ed:81:5a:06:83:33:ba:2d:79:a8:18:57:
    67:f9:ab:ea:7d:21:a7:39:bf:dc:e2:b7:3c:54:e6:
    40:9b:14:ce:b9:37:cd:44:db:2e:e6:03:af:46:c9:
    d1:94:52:1b:6d:fe:7c:fb:fd:1c:6e:0a:4d:cb:d4:
    32:a2:e7:fb:30:35:cf:14:0e:b2:d3:9f:d4:95:17:
    23:e0:88:b4:05:38:03:f4:e1:dc:e3:66:37:3f:36:
    70:76:ca:82:b5:34:43:e5:c1:c2:2f:40:d5:9b:7b:
    f0:5d
exponent1:
    7f:8a:27:a6:22:88:e6:e1:e5:26:87:b1:ad:1f:d2:
    98:f1:a8:a4:4b:a9:bd:7d:1b:7d:c5:d5:c4:19:4a:
    2f:7d:96:73:07:2c:9d:b1:b7:5c:a5:d2:41:7f:59:
    94:bd:6a:8b:0b:bd:ba:c9:d6:98:5c:f7:22:7e:6d:
    38:f0:7d:91:8d:ab:e9:05:8b:c6:20:6c:14:5e:63:
    2c:c2:26:76:f4:ce:bf:c4:95:44:e1:7a:53:24:df:
    b7:d3:64:f2:0a:1a:0e:1d:5d:64:4b:93:19:10:73:
    ee:66:8b:74:c7:b8:cf:70:aa:10:82:e8:72:2c:3e:
    5e:62:ac:2b:95:ed:11:fd:5c:fd:47:1f:eb:b2:fe:
    5e:25:7a:4d:bf:9c:a0:0f:2f:e7:eb:7c:2a:16:28:
    bb:0f:57:fb:62:9c:a6:9b:ba:93:13:2f:e4:d0:d2:
    f7:b4:78:52:3a:2f:85:55:13:0b:66:f3:8a:39:ac:
    12:f8:d7:08:5e:89:58:e4:ed:3b:31:a1:f4:d3:ae:
    43:5a:7f:f5:d3:68:88:d3:40:ba:ed:01:19:30:4e:
    b6:d0:cc:eb:0f:58:fc:66:7c:fc:08:89:ad:1c:76:
    0b:8d:40:61:64:f8:dc:25:cc:86:1c:70:3f:dd:dc:
    2a:f2:1c:67:d5:de:52:42:f3:7c:70:6d:09:05:68:
    05
exponent2:
    0c:dd:c7:08:4e:55:2c:28:c5:ef:3a:82:3a:5b:9f:
    74:aa:1d:df:d9:20:95:07:7d:dc:99:c3:63:bc:3b:
    29:5e:b4:b0:35:f7:09:9a:1d:04:12:7d:e2:a5:62:
    8c:2a:a0:59:55:77:ca:5f:b0:1c:bc:93:76:d3:37:
    f3:ea:f7:54:91:50:cd:01:ac:bd:63:58:fa:d7:87:
    17:08:8f:c2:af:dd:5d:2d:ad:73:95:35:16:14:35:
    4c:50:57:af:15:89:38:a6:94:9f:7e:81:32:99:6f:
    78:3e:23:21:1c:96:45:f9:60:5c:36:cd:29:14:9a:
    b4:01:37:6b:c7:b2:86:68:d2:d5:d4:47:1d:08:ea:
    9c:91:a1:0d:ff:b1:2c:ce:5d:f6:08:1a:85:78:ff:
    6e:2c:8a:70:7e:a7:ee:2a:ac:df:43:37:6d:e2:75:
    99:5c:50:8b:df:ce:74:5e:1e:17:d4:88:5e:6d:d8:
    c6:a8:d8:b0:8c:fb:10:ca:a1:d8:9d:d2:2a:0c:42:
    6d:98:af:1b:1e:32:67:f7:f5:14:b6:e8:79:b3:e3:
    14:ee:df:f9:96:c6:65:e5:cc:a0:d9:a1:60:df:2d:
    28:37:08:2d:5d:69:6d:69:48:88:b4:40:f5:1c:d6:
    3f:2f:d7:93:6a:6b:9f:06:37:9e:c2:e7:24:d6:77:
    c9
coefficient:
    00:e4:37:77:5e:05:2f:ba:4a:a9:f2:c7:5a:88:93:
    2d:47:e7:92:7f:bf:18:16:71:4e:5e:a6:39:9c:fb:
    06:72:fc:44:3d:7e:42:a1:f6:1c:de:ed:c1:9b:0f:
    9e:40:f8:84:34:d6:22:e1:8e:ad:76:d4:52:ae:0d:
    6a:37:a7:0c:a1:be:eb:83:be:b8:ea:79:67:8b:d3:
    e8:59:6b:ab:e2:06:2f:1c:a2:2a:71:63:1e:1a:a1:
    26:12:46:d4:f6:3a:26:85:38:bc:ea:09:da:bf:a6:
    22:05:f2:d2:8f:f0:e0:7f:c4:03:52:8a:5b:cb:c0:
    65:e2:9c:7b:3f:f0:88:c5:74:0c:f4:28:9e:07:7f:
    fa:b7:2f:3c:d1:4a:4f:39:91:9a:d1:19:15:e3:6b:
    47:82:f4:c9:ee:36:4e:a0:0f:79:84:de:6c:ac:35:
    05:db:13:a7:1d:a9:34:34:df:30:5b:2e:b7:13:c5:
    24:80:d1:e1:54:bb:fc:2e:07:cb:6c:89:76:b5:55:
    4c:38:94:c3:d8:8c:94:7d:ca:80:b9:27:17:26:43:
    d1:7d:e3:02:51:eb:e7:f3:b2:41:9e:0b:f9:08:19:
    d2:b6:2c:aa:49:db:a1:e0:e4:57:34:75:d6:b9:5a:
    90:ac:a4:e9:36:95:82:73:41:1c:93:a1:41:65:7f:
    74:51

† ruby を使って 10 進数表示してみる

OpenSSL のコマンド単体では大きな数の10進表現は無理なようなので、Ruby の OpenSSL::PKey::RSA^*1を使って簡単なスクリプトを書いてみました。

rsa_dump.rb

require 'openssl'

def to_h(n)
  n.to_s(16).gsub( /(.{2})(?=.)/, '\1:')
end

k = OpenSSL::PKey::RSA.new(File.read(ARGV[0]))

puts <<"EOS"
modulus: #{k.n.to_i} ( #{to_h(k.n)} )
publicExponent: #{k.e.to_i} ( #{to_h(k.e)} )
privateExponent: #{k.d.to_i} ( #{to_h(k.d)} )
prime1: #{k.p.to_s} ( #{to_h(k.p)} )
prime2: #{k.q.to_s} ( #{to_h(k.q)} )
exponent1: #{k.dmp1.to_i} ( #{to_h(k.dmp1)} )
exponent2: #{k.dmq1.to_i} ( #{to_h(k.dmq1)} )
coefficient: #{k.iqmp.to_i} ( #{to_h(k.iqmp)} )
EOS

これに秘密鍵を食わせると秘密鍵に含まれる値を10進表現として取り出すことができます。具体的な出力を確認してみると、prime1, prime2 共に 10進で 617 桁であることが分かります。また、これらを掛合わせた値 modulus は 10進で 1233 桁になっています。任意精度演算が可能な計算機を使うと prime1 * prime2 == modulus であることが確認できます。

$ ruby rsa_dump.rb id_rsa
modulus: 887153319317956627464195922141130043378803095776919540308803921849592282862879218339693248100317798454505964133548960908960638673874867346370837503740750148818257618484107020482052685280189700919464696188390164261869726720586224546950503256452211064830663895120017797965758903589481942299249373760780159646270795328403114142929673015396554286491784674946375800942213291357707858557165939046412231277134429235791484051815463532242176000819792249307849953555879287011517210271297817794569305241859306085173312201118089521467747561131899553663783619813480244623162602954139901271835889030169281355450419829618289884693042638913327755165482624230062991497460271776095949773375523255211431534724231665640071912409294116111501857113739146326662119315438534592052708523462188343472458816149932750539949578970970916115362604494235168893427179932798301542012890618593755491593429902839189655204343625349883445285816797291501164212645809068135983861969473370760397947799347366862533906582897973854202964004549929535591947796206321145195553703454866056620625159475167270748112642302598940097346174634047862520608620492033905862741155828072304109939406528477613614206429683331709976266035641201579647653895192907968308772489160543760376304426257 ( D9:75:60:E5:03:BE:F0:42:7B:B0:7E:8E:DC:76:92:8C:DA:6C:85:64:E6:42:67:A2:C5:B7:4F:7E:F3:1E:4E:52:CD:1F:6D:81:5A:F6:A6:1E:A5:9E:B5:37:AF:6B:98:5F:13:1F:CF:AE:F0:69:9B:A2:15:CE:BC:F9:55:62:B8:D2:C2:4A:F7:6A:96:14:3F:10:DA:B6:38:2A:A9:95:19:D3:B8:42:56:A8:19:BC:FC:80:25:70:25:2A:67:08:BA:C3:26:EB:EF:A9:D4:CD:99:3C:51:BA:56:A6:B9:A2:E0:2F:F5:AF:5C:62:2D:83:97:CF:FE:D0:36:55:52:FC:F7:FF:5D:19:D7:F5:C2:C1:93:06:52:FB:4B:A4:21:FB:25:F2:B1:9F:DF:09:66:8F:BC:4A:3A:F1:97:6F:FA:CA:A2:67:95:4F:D4:A8:C4:84:6A:AF:8E:EE:1D:01:7D:1C:2E:69:ED:C8:AC:D4:D7:8D:D4:95:86:51:42:6B:B3:D4:0E:86:3A:62:BD:D4:CF:A0:C0:03:4D:35:E8:3A:C3:F6:2E:FA:29:B5:EA:46:83:5A:06:B5:36:84:B6:90:4A:06:2A:30:1F:AA:A9:E2:AE:E8:27:25:C8:49:2B:3F:73:BD:19:8F:C7:52:32:47:91:61:CF:2B:CA:CC:39:56:32:5B:A8:2E:96:D6:4F:DC:CB:97:0E:0F:E7:0E:A2:D1:19:9F:C0:91:9F:E0:19:EB:A5:57:C3:58:AC:DD:D1:05:6E:CE:61:52:20:D7:7F:62:09:07:5F:4E:95:F3:51:04:C2:F3:CC:5E:60:7B:8A:A6:C3:51:87:9A:1F:DA:0A:F9:81:22:E3:00:C8:22:3E:53:EA:17:35:77:63:FB:34:3F:A8:5F:75:AE:D5:1B:3F:8F:7F:89:3F:4D:F4:40:15:56:9B:7B:DB:95:73:77:85:80:7E:EC:F8:57:E7:08:D2:C9:51:54:40:B0:04:E9:EE:7C:B4:6D:06:61:31:3A:78:BD:2D:B9:98:B1:EA:5A:EC:4D:FB:99:3C:28:0E:C7:C9:E4:C9:31:9B:8D:43:45:88:F8:92:DA:C1:70:84:F7:54:9B:99:07:80:3D:F4:69:82:EB:0B:D5:A6:74:FF:BF:D8:82:24:B5:07:B5:AC:43:DB:10:04:FD:99:93:58:BA:28:D6:57:EC:AD:69:07:87:D9:C9:67:D6:FE:32:D5:3B:65:B1:59:75:36:2E:A2:D0:8D:91:2C:75:4B:F0:67:69:D4:68:D0:70:91:DA:F7:D0:C0:F1:E8:5A:AC:09:63:4C:7B:B0:30:FA:32:4B:F9:FF:15:F2:A2:C7:4B:E0:1A:A1:0B:FD:EA:AE:DD:11 )
publicExponent: 65537 ( 01:00:01 )
privateExponent: 147468578980573103736743372076925564071725604244835153762364922480880393205490123206625543506795582592480399976667872806845250739478352760598805312506702048022126409444525411311648106465697035290242891805794016349067073605658884755397390519489607204178788355790430960999725002604693780298274603319497979144398950887401369081176676653336742029648008029797912903786631545478903053406804793322422674939852334896237429654400989665688790535925215019972835457742004500552412659851618451059005416959958729885284313641438888982511705478294259940760383581095382055707840355777383769236543939684371639701028073815155738743058811911808042974127417907059195629010420642082732089568714601530794116447813631652408652958878790019862125712772908055273255532990588261730429431607835698758280348180841208306504129933597022879294454468213360271988737896765685235079980117672836355968461937637459021673637171457129961183061373846589466738763091626266287207781629415319356777800480568955275066468992629817023271674417533869246693115614496369030623700291634493784796621702721990637900941052662763050930896765438179210907446436402733145484009922607268276625937938816133806260291678001044582360436151421726711506564248681799891758774862415937856446928031265 ( 24:25:BC:DB:94:C9:D2:8A:5A:A1:36:08:6D:61:50:6F:15:87:E0:8C:AB:31:F2:A4:3E:15:80:A3:6B:AA:67:FE:9F:96:4F:CF:A6:DD:45:33:E6:F8:00:CB:A6:DF:4D:CE:D0:11:03:A6:E2:D3:CB:49:20:32:E3:7A:D6:D4:B8:40:60:87:B1:A5:08:19:89:7A:35:C1:00:89:26:F3:5F:FD:AF:AD:67:56:3E:21:0A:20:0B:19:1F:34:73:38:22:43:7A:5E:D1:D1:FB:4B:C9:E3:1A:FA:CC:98:9E:57:12:C9:26:1F:E6:4D:73:9F:62:46:F2:E6:8D:81:95:6F:F3:91:81:5A:60:69:BB:DC:0D:A2:89:7C:B8:50:2A:4D:E3:3A:FA:87:C6:BD:7B:4C:FD:51:A1:86:B3:53:AB:EE:AF:1C:64:D7:86:54:1F:86:9B:6D:5F:66:F9:6F:5B:EA:A9:54:76:70:5B:3A:6E:32:67:5A:0B:9F:CA:58:B8:E5:D1:1C:42:EC:52:FD:E3:1D:A7:8D:64:F2:47:0E:2D:B0:E8:92:31:08:CA:4E:BA:5A:E6:AE:8C:C7:76:C9:79:64:98:F2:B2:F0:DD:98:8E:D6:8C:5B:5D:75:BC:38:A9:F6:8F:1F:35:54:D6:82:CB:09:E5:34:34:5B:CA:54:27:BE:D0:AF:CF:4E:36:BE:E5:51:27:9B:A5:95:E2:EB:CE:06:2D:F9:2D:15:59:BB:BD:72:C0:A7:1C:8B:FD:6F:CA:9F:FA:E4:1D:CB:A2:F4:3E:C5:96:51:D7:BF:48:BC:31:66:CF:90:AD:73:49:7E:60:C8:EB:9E:4B:3B:52:81:E7:F3:20:8D:9F:6C:FE:82:20:6E:0C:CC:5E:8A:90:15:CF:A9:49:50:D2:86:97:19:7B:BB:38:20:39:89:9A:A3:58:A7:1C:79:01:88:D6:B7:97:EE:B6:87:0B:57:FE:4D:F2:2C:39:F8:84:A0:F3:75:C9:43:ED:46:91:15:70:C1:A3:41:74:B3:9F:AB:16:64:A7:BD:CD:FE:42:53:34:D3:20:49:CC:6A:F5:94:A1:6A:1D:CE:98:C7:3C:EF:DD:90:EC:33:42:B0:62:9F:6B:7C:A7:6A:31:14:F2:1E:EB:8F:EC:27:30:AF:EE:BB:7A:C6:C8:75:01:10:4B:D5:67:E6:15:5F:49:12:7E:15:15:A4:74:EA:C0:AB:46:3F:0C:55:07:8E:C4:9E:F5:1E:E9:8C:93:39:E8:10:54:E8:9C:3B:CC:A9:40:63:B6:5A:A3:99:A4:22:5E:87:57:3C:34:F9:3C:20:F3:DA:C5:C4:1D:62:28:5C:67:99:9F:F9:37:D6:A1:50:8E:21 )
prime1: 31720166957397019562536359576195634670385815683413551851432819901172721554341254487492090525224012507111096587265694200074580556701432764155935173355262321217273424812781466916711213516333112139501227366719828188141473356546717555914550064801745405438318787485987538659089333759195833583845990820696586226108301212916037544088762637057848914536136275983436637026653565067393995017607963644112128321812669442735717965218687157577701897875321766858225859746250695874328892870575811995678439394944366142822339452607290598798153398455306844448972792128953931452910617504345141340711498863488937788831077038679103365063749 ( FB:45:A9:E2:F1:31:90:62:F7:CA:B8:8F:CB:4F:94:73:21:7B:2B:A9:15:8C:A1:A6:4E:07:AD:45:17:D6:04:91:CD:C0:56:6F:AC:22:4C:EE:06:29:E0:0B:4A:51:9D:1A:8F:CA:42:37:F1:E9:77:E3:9D:36:7D:8B:F5:4C:F8:28:89:16:A3:7C:21:9C:D5:6F:AD:F3:83:91:6F:F1:C9:66:8B:40:76:5E:78:1F:15:B6:EE:1D:2C:20:E1:DD:DE:9D:CF:AE:73:A6:46:51:BF:DC:25:35:3C:3F:C7:03:9F:1E:7E:8B:A8:17:47:86:71:5C:EA:7B:A1:15:10:67:9D:83:46:3E:89:68:DC:7D:B4:0D:2C:89:F6:C3:B0:EA:69:72:7F:A3:30:88:B1:E2:EF:4B:04:CC:D1:E9:07:A1:22:B4:D9:52:1F:24:83:C9:5B:0C:86:C1:C7:17:FE:08:64:5E:47:CF:86:A3:D1:37:7A:98:67:62:04:47:EC:FA:1A:9E:EF:CC:2F:5E:25:F7:36:C4:17:55:DC:32:21:B4:98:D2:FF:8B:92:50:E9:4D:D8:46:56:63:84:5D:19:8E:31:31:45:02:14:84:33:F8:E8:0D:39:F6:49:88:69:10:A9:85:D7:EC:2B:B2:3A:85:17:7D:56:D3:A6:08:46:C2:24:45 )
prime2: 27968116325159377002575942957519338470295343925207961486561472243064977998054758141207601759298529186901056285783602811060839143886458443726252028027966088216340272208248329395116146054225793665538035535906001130665223711122774417835354329753432786257001426886899250273975992900925719440794215325781477351625087200394837415306175045738059707937798237761598538294083557102014061120050356857100645798310620936902148386221324926744759452962880161856292715330397990059424357537113417769163970107010399313680676306386525790001662879153823119974484506912431631750516640577765807423779814925743718805563889148619187174502493 ( DD:8C:D7:1C:35:1A:21:7E:30:B2:AD:A2:6D:E5:7C:74:55:BA:72:D1:14:75:4F:6D:E9:A4:CA:B1:D0:D7:9D:A7:5A:31:6A:16:D7:0C:26:15:38:C4:36:91:5D:DE:2D:C5:E1:FD:1D:B8:20:FD:BD:DC:3A:E1:A4:B5:D1:A6:A0:0D:7E:F2:D6:99:34:CE:18:E9:69:A2:C5:1D:B4:DA:AF:A9:E8:64:D4:2F:F9:B8:BF:3E:E8:F4:54:DC:59:D2:E1:C0:A6:CC:06:53:E2:73:7F:08:9D:3C:C7:52:AE:97:F8:A3:8E:E3:2F:29:64:36:77:71:71:EF:D1:85:86:01:C1:89:6E:E4:5C:E2:A0:9F:CA:8F:26:72:DB:DC:F7:D3:D8:5D:EC:C4:71:60:27:12:A4:A7:ED:81:5A:06:83:33:BA:2D:79:A8:18:57:67:F9:AB:EA:7D:21:A7:39:BF:DC:E2:B7:3C:54:E6:40:9B:14:CE:B9:37:CD:44:DB:2E:E6:03:AF:46:C9:D1:94:52:1B:6D:FE:7C:FB:FD:1C:6E:0A:4D:CB:D4:32:A2:E7:FB:30:35:CF:14:0E:B2:D3:9F:D4:95:17:23:E0:88:B4:05:38:03:F4:E1:DC:E3:66:37:3F:36:70:76:CA:82:B5:34:43:E5:C1:C2:2F:40:D5:9B:7B:F0:5D )
exponent1: 16100391440526906262840410780202752449919650864530750604054392999565292621040966637569989339175988770451052504316543594694308897549066342671272465045131774650847620068010062971823527436727664911737008534933473986885669334963861017402345360721883224924938194694926155812664703716978949359394492952843003811762708696578299112014780797438536157316394910059798582948435705661944569392262827267366372409861581229116432825930384794800223593280476350375190857446313218460709990102380401697304473602283051402123764009505798583533280021966458369784931869480898614977509982014465738845213665710880258717143991603089252993558533 ( 7F:8A:27:A6:22:88:E6:E1:E5:26:87:B1:AD:1F:D2:98:F1:A8:A4:4B:A9:BD:7D:1B:7D:C5:D5:C4:19:4A:2F:7D:96:73:07:2C:9D:B1:B7:5C:A5:D2:41:7F:59:94:BD:6A:8B:0B:BD:BA:C9:D6:98:5C:F7:22:7E:6D:38:F0:7D:91:8D:AB:E9:05:8B:C6:20:6C:14:5E:63:2C:C2:26:76:F4:CE:BF:C4:95:44:E1:7A:53:24:DF:B7:D3:64:F2:0A:1A:0E:1D:5D:64:4B:93:19:10:73:EE:66:8B:74:C7:B8:CF:70:AA:10:82:E8:72:2C:3E:5E:62:AC:2B:95:ED:11:FD:5C:FD:47:1F:EB:B2:FE:5E:25:7A:4D:BF:9C:A0:0F:2F:E7:EB:7C:2A:16:28:BB:0F:57:FB:62:9C:A6:9B:BA:93:13:2F:E4:D0:D2:F7:B4:78:52:3A:2F:85:55:13:0B:66:F3:8A:39:AC:12:F8:D7:08:5E:89:58:E4:ED:3B:31:A1:F4:D3:AE:43:5A:7F:F5:D3:68:88:D3:40:BA:ED:01:19:30:4E:B6:D0:CC:EB:0F:58:FC:66:7C:FC:08:89:AD:1C:76:0B:8D:40:61:64:F8:DC:25:CC:86:1C:70:3F:DD:DC:2A:F2:1C:67:D5:DE:52:42:F3:7C:70:6D:09:05:68:05 )
exponent2: 1624222206288914489094771486279790076108825228181660762895051091095187546891014380967028278619561500913154709914893759233677980097225396902850530519774157067784474053200377522282253564892860074325003635345808326644671581618525099322235661977837941689338044627180654386724333261835654488177102759203569019031769563524463298635203049026947453322722433021356547244263271408979134177989710517694204158084291671664091684971212638222539244670594044524849322894662476923969194573847653509154188782325733246683074507867420192513333367686336737943801028934933164326143496559790296520361108619671034587698188231070153140762569 ( 0C:DD:C7:08:4E:55:2C:28:C5:EF:3A:82:3A:5B:9F:74:AA:1D:DF:D9:20:95:07:7D:DC:99:C3:63:BC:3B:29:5E:B4:B0:35:F7:09:9A:1D:04:12:7D:E2:A5:62:8C:2A:A0:59:55:77:CA:5F:B0:1C:BC:93:76:D3:37:F3:EA:F7:54:91:50:CD:01:AC:BD:63:58:FA:D7:87:17:08:8F:C2:AF:DD:5D:2D:AD:73:95:35:16:14:35:4C:50:57:AF:15:89:38:A6:94:9F:7E:81:32:99:6F:78:3E:23:21:1C:96:45:F9:60:5C:36:CD:29:14:9A:B4:01:37:6B:C7:B2:86:68:D2:D5:D4:47:1D:08:EA:9C:91:A1:0D:FF:B1:2C:CE:5D:F6:08:1A:85:78:FF:6E:2C:8A:70:7E:A7:EE:2A:AC:DF:43:37:6D:E2:75:99:5C:50:8B:DF:CE:74:5E:1E:17:D4:88:5E:6D:D8:C6:A8:D8:B0:8C:FB:10:CA:A1:D8:9D:D2:2A:0C:42:6D:98:AF:1B:1E:32:67:F7:F5:14:B6:E8:79:B3:E3:14:EE:DF:F9:96:C6:65:E5:CC:A0:D9:A1:60:DF:2D:28:37:08:2D:5D:69:6D:69:48:88:B4:40:F5:1C:D6:3F:2F:D7:93:6A:6B:9F:06:37:9E:C2:E7:24:D6:77:C9 )
coefficient: 28809684973535337606438418494311224749466178540761610608827490675553484780976386459460256993111303065623189622302100993383301604920324500779219344848819135516998320284779838491469032785481768918128844366563256417933555574769513028154028340296159483186176279264925214447634745200464723488271988625990543977626839656382789941763494963678149493813821187850420772208531660241199544111586184678574272551048610028819168130715845505822547068088191204934605735269934766301036622918961995543608099477680689198601710836303728276110166959200740341385837595587738276554250938511456168461012436702837917036842064114611324268278865 ( E4:37:77:5E:05:2F:BA:4A:A9:F2:C7:5A:88:93:2D:47:E7:92:7F:BF:18:16:71:4E:5E:A6:39:9C:FB:06:72:FC:44:3D:7E:42:A1:F6:1C:DE:ED:C1:9B:0F:9E:40:F8:84:34:D6:22:E1:8E:AD:76:D4:52:AE:0D:6A:37:A7:0C:A1:BE:EB:83:BE:B8:EA:79:67:8B:D3:E8:59:6B:AB:E2:06:2F:1C:A2:2A:71:63:1E:1A:A1:26:12:46:D4:F6:3A:26:85:38:BC:EA:09:DA:BF:A6:22:05:F2:D2:8F:F0:E0:7F:C4:03:52:8A:5B:CB:C0:65:E2:9C:7B:3F:F0:88:C5:74:0C:F4:28:9E:07:7F:FA:B7:2F:3C:D1:4A:4F:39:91:9A:D1:19:15:E3:6B:47:82:F4:C9:EE:36:4E:A0:0F:79:84:DE:6C:AC:35:05:DB:13:A7:1D:A9:34:34:DF:30:5B:2E:B7:13:C5:24:80:D1:E1:54:BB:FC:2E:07:CB:6C:89:76:B5:55:4C:38:94:C3:D8:8C:94:7D:CA:80:B9:27:17:26:43:D1:7D:E3:02:51:EB:E7:F3:B2:41:9E:0B:F9:08:19:D2:B6:2C:AA:49:DB:A1:E0:E4:57:34:75:D6:B9:5A:90:AC:A4:E9:36:95:82:73:41:1C:93:A1:41:65:7F:74:51 )

こうやって実際に値が確認できると、RSA は単純ですが興味深いですね。

† 参考

今回使った秘密鍵のファイルは以下になります。

$ cat id_rsa
-----BEGIN RSA PRIVATE KEY-----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-----END RSA PRIVATE KEY-----