incubator-milagro-crypto-c
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Incorrect RSA private key generation and potentials for fault attack
Hi,
I recently run into a rather interesting problem. It appears to me that the RSA key generation in milagro might sometimes be incorrect, and I have no ideas to why that would be the case.
Here's the source code I prepared to illustrate the problem. For simplicity I just replaced the source code of test/test_rsa_2048.c with this. The build target I used was LINUX_64BIT_NIST256_RSA2048.
#include <stdio.h>
#include <time.h>
#include "rsa_2048.h"
#include "randapi.h"
void print_keypair(rsa_private_key_2048 *priv, rsa_public_key_2048 *pub)
{
printf(" priv->p = ");
FF_2048_output(priv->p,FFLEN_2048/2);
printf("\n");
printf(" priv->q = ");
FF_2048_output(priv->q,FFLEN_2048/2);
printf("\n");
printf("\n");
printf(" pub->n = ");
FF_2048_output(pub->n,FFLEN_2048);
printf("\n");
printf("\n");
printf("priv->dp = ");
FF_2048_output(priv->dp,FFLEN_2048/2);
printf("\n");
printf("priv->dq = ");
FF_2048_output(priv->dq,FFLEN_2048/2);
printf("\n");
printf(" priv->c = ");
FF_2048_output(priv->c,FFLEN_2048/2);
printf("\n");
}
int main(int argc, char *argv[])
{
char m[RFS_2048],ml[RFS_2048],c[RFS_2048],s[RFS_2048];
char p[RFS_2048/2], q[RFS_2048/2];
rsa_public_key_2048 pub;
rsa_private_key_2048 priv;
octet M= {0,sizeof(m),m};
octet ML= {0,sizeof(ml),ml};
octet C= {0,sizeof(c),c};
octet S= {0,sizeof(s),s};
octet P= {sizeof(p), sizeof(p), p};
octet Q= {sizeof(q), sizeof(q), q};
// somehow this prime number leads to incorrect dp, which is quite weird
OCT_fromHex(&P, "f55d1f9bde519d3d30e9a0820a0850f3a6d05efa82dcc7ef948a09834224ad6fb77ee792f53b1ee6efc0270e5e8fdd59fab40ada833df0ea69eb2f0b37d717a3fd0329280cd47525fd22970c8fd860c7be749482c42a6e16a4b5b7edf27e01214f55fd5839a11c86be6bbd8b01e5ddeef743bca59e896c1751145d4b6a9da9ed");
OCT_fromHex(&Q,"f34e8af7a6b3b2d2906890db77f56b8c4c5a5af42bec8cacd79fc479606d85eeb641ec95aef73d0e7e24f0da9115a524d9787d573ed3a1868bff2765cc5b9e1238a7931bdd5afd51a0bd6bf93f26909fff6d93d335e6785959c14f851205ad4a8ce995e6f92e490e7a5feab2b9fcda0080bedc17fa816227f2d6034044c0c363");
// use our choice of P and Q
RSA_2048_KEY_PAIR(NULL,3,&priv,&pub,&P,&Q);
print_keypair(&priv, &pub);
// ======
printf("\n===\n\n");
OCT_fromHex(&P, "db59a11194cb81be97715d9ea436484c7fd39cee1a71dd1e896e08cc2d91d95a2d6d561f48db72acea57450ecfd4dc5399c1f8ce7abbb8b99c92f9c7be7ab2df6990b473c5a464eaa4d34747d022e47c17b549905db305986af5970442ad54714ee7fce2c8c443da7b22f4354e105c009a8d36de733d9fb02bfbcd57803bdcfb");
OCT_fromHex(&Q, "f6457c323b66f43fa7a20f03b01a8c09c257f7fa65c7ce6a01acd816bca4de5da7692fd82ac036ca18bcc1d36e03a9ca6c2c1cf9007c106932c946cb51acb151c5c2912369553a70ddea742bc5673baaa962f8028c5ec7f4699bffd6c7beab3c278424a15548a4fce75946784036d2a9a118698224b3f218e66d980786252c93");
// use our choice of P and Q
RSA_2048_KEY_PAIR(NULL,3,&priv,&pub,&P,&Q);
print_keypair(&priv, &pub);
// === clean up ===
RSA_2048_PRIVATE_KEY_KILL(&priv);
OCT_clear(&M);
OCT_clear(&ML); /* clean up afterwards */
OCT_clear(&C);
return 0;
}
Which produces the following output:
priv->p = f55d1f9bde519d3d30e9a0820a0850f3a6d05efa82dcc7ef948a09834224ad6fb77ee792f53b1ee6efc0270e5e8fdd59fab40ada833df0ea69eb2f0b37d717a3fd0329280cd47525fd22970c8fd860c7be749482c42a6e16a4b5b7edf27e01214f55fd5839a11c86be6bbd8b01e5ddeef743bca59e896c1751145d4b6a9da9ed priv->q = f34e8af7a6b3b2d2906890db77f56b8c4c5a5af42bec8cacd79fc479606d85eeb641ec95aef73d0e7e24f0da9115a524d9787d573ed3a1868bff2765cc5b9e1238a7931bdd5afd51a0bd6bf93f26909fff6d93d335e6785959c14f851205ad4a8ce995e6f92e490e7a5feab2b9fcda0080bedc17fa816227f2d6034044c0c363
pub->n = e932ac92252f585b3a80a4dd76a897c8b7652952fe788f6ec8dd640587a1ee5647670a8ad4c2be0f9fa6e49c605adf77b5174230af7bd50e5d6d6d6d28ccf0a886a514cc72e51d209cc772a52ef419f6a953f3135929588ebe9b351fca61ced78f346fe00dbb6306e5c2a4c6dfc3779af85ab417371cf34d8387b9b30ae46d7a 5ff5a655b8d8455f1b94ae736989d60a6f2fd5cadbffbd504c5a756a2e6bb5cecc13bca7503f6df8b52ace5c410997e98809db4dc30d943de4e812a47553dce54844a78e36401d13f77dc650619fed88d8b3926e3d8e319c80c744779ac5d6abe252896950917476ece5e8fc27d5f053d6018d91b502c4787558a002b9283da7
priv->dp = 4c8b6ace1ba4747563b3d9f6c29e01fb968c96e116707fb4556b811f39b047aac0d7fc303650ff6a0a14e969656094a08dab8231a71c76613ae1063d87aea343b8a2146fc5b850693e1d7167ab97e063b8c383462e8394f47ff0fdb4c6d4e5872c49c2b459841670b808f68173248883f16d95c698b0dd39a5e167e840047571 priv->dq = a2345ca519cd21e1b59b0b3cfaa39d0832e6e74d729db31de5152da6404903f479814863c9fa28b45418a091b60e6e1890fafe3a29e26baf07ff6f9932e7beb6d06fb767e8e7538bc07e47fb7f6f0b1554f3b7e223eefae63bd6350361591e31b34663ef50c98609a6ea9c7726a89155ab29e80ffc56416ff73957802dd5d797 priv->c = 6f1e5e6d113bffba6e9fe60cbfd0b959f0b78fb8f71db4ab4f63147abec85beb2a982acde258672046843ce3b9f93a33ff1b25602762dfd8dbddf36e3b9815a1bf14a9dbba3d28df3dfb830fb976662cc1b41026207aa16686f49c23f93f1312ba29ef76a0c13fb03e727247b632dda5a06bf6d385da9831af4ec18d1d23df95
===
priv->p = db59a11194cb81be97715d9ea436484c7fd39cee1a71dd1e896e08cc2d91d95a2d6d561f48db72acea57450ecfd4dc5399c1f8ce7abbb8b99c92f9c7be7ab2df6990b473c5a464eaa4d34747d022e47c17b549905db305986af5970442ad54714ee7fce2c8c443da7b22f4354e105c009a8d36de733d9fb02bfbcd57803bdcfb priv->q = f6457c323b66f43fa7a20f03b01a8c09c257f7fa65c7ce6a01acd816bca4de5da7692fd82ac036ca18bcc1d36e03a9ca6c2c1cf9007c106932c946cb51acb151c5c2912369553a70ddea742bc5673baaa962f8028c5ec7f4699bffd6c7beab3c278424a15548a4fce75946784036d2a9a118698224b3f218e66d980786252c93
pub->n = d303aa59c3248225f2ff52e32e33cdcb797c3b501fed5e51c0189f27ef7cd614c7c4cbf2079e663b24f7ae2e9683b4878ce662d6c944196874f4a48422353303c9ad8ffd2824b4f70996ee558a89e1e98679305209a124aaf62eecd3187d2d8ee2cf5eb1b5a1b7347958aa53205b1f726856903d6a953ed72a74673c92f46f52 92b4f1c59cdb645a6afc4d0381dec80444775658c54995bf0cba58e5f90f152a557172bc040c432fe63ede5b1cf14f2f1a11d30d3a96baa24c3a3982461292be8e9db35fd088a8cbaf0e26326e630ee690368cd0a8868502e9ddb17dde45179ce7bb22b10f9deb2e3d8ddcd30ca97c9bae84ca7c3c9a13afa95de4783ea20821
priv->dp = 923bc0b66332567f0fa0e9146d7985885537bdf411a13e145b9eb0881e613b91739e396a30924c7346e4d8b48a8de837bbd6a5defc7d25d1130ca68529a721ea4660784d2e6d989c6de22f85356c9852ba78dbb593ccae659ca3ba02d71e384b89effdec85d82d3c52174d78deb592ab11b379e9a22915201d5288e50027e8a7** priv->dq = a42e52cc2799f82a6fc15f57cabc5d5bd6e54ffc43da899c011de5647dc33ee91a461fe571d579dc107dd68cf40271319d72bdfb55a80af0cc862f32367320e12e81b6179b8e26f5e946f81d2e44d271c641faac5d94854d9bbd5539da7f1cd2c502c31638db18a89a3b84502acf371bc0baf1016dcd4c10999e655a596e1db7 priv->c = b163cc8298e63ea294f20288143304adc9a555ca170e55942954773801405652d8f998da549541782bb565ff49b32ae5f6d814bf38a878c598c6307fc294a0151f2681b8a39d6ace0a732feb6cb514a03de8d92290a8f1499231a945d698c8461cde13156d9bfc5b09ca3bf0d27a770f3ab3fedb375bd4135c8dcb57f9d60881
The primes P and Q were all copied from RSA private keys generated by OpenSSL. If we use OpenSSL to check the numbers of the 2 private keys:
$ openssl rsa -in privkey.pem -noout -text
RSA Private-Key: (2048 bit, 2 primes)
modulus:
00:e9:32:ac:92:25:2f:58:5b:3a:80:a4:dd:76:a8:
97:c8:b7:65:29:52:fe:78:8f:6e:c8:dd:64:05:87:
a1:ee:56:47:67:0a:8a:d4:c2:be:0f:9f:a6:e4:9c:
60:5a:df:77:b5:17:42:30:af:7b:d5:0e:5d:6d:6d:
6d:28:cc:f0:a8:86:a5:14:cc:72:e5:1d:20:9c:c7:
72:a5:2e:f4:19:f6:a9:53:f3:13:59:29:58:8e:be:
9b:35:1f:ca:61:ce:d7:8f:34:6f:e0:0d:bb:63:06:
e5:c2:a4:c6:df:c3:77:9a:f8:5a:b4:17:37:1c:f3:
4d:83:87:b9:b3:0a:e4:6d:7a:5f:f5:a6:55:b8:d8:
45:5f:1b:94:ae:73:69:89:d6:0a:6f:2f:d5:ca:db:
ff:bd:50:4c:5a:75:6a:2e:6b:b5:ce:cc:13:bc:a7:
50:3f:6d:f8:b5:2a:ce:5c:41:09:97:e9:88:09:db:
4d:c3:0d:94:3d:e4:e8:12:a4:75:53:dc:e5:48:44:
a7:8e:36:40:1d:13:f7:7d:c6:50:61:9f:ed:88:d8:
b3:92:6e:3d:8e:31:9c:80:c7:44:77:9a:c5:d6:ab:
e2:52:89:69:50:91:74:76:ec:e5:e8:fc:27:d5:f0:
53:d6:01:8d:91:b5:02:c4:78:75:58:a0:02:b9:28:
3d:a7
publicExponent: 3 (0x3)
privateExponent:
00:9b:77:1d:b6:c3:74:e5:92:27:00:6d:e8:f9:c5:
ba:85:cf:98:c6:37:54:50:5f:9f:30:93:98:03:af:
c1:49:8e:da:44:b1:b1:e3:2c:7e:b5:15:19:ed:bd:
95:91:ea:4f:ce:0f:81:75:ca:52:8e:09:93:9e:48:
f3:70:88:a0:70:59:c3:63:32:f7:43:68:c0:68:84:
f7:18:c9:f8:11:4f:1b:8d:4c:b7:90:c6:3b:09:d4:
67:78:bf:dc:41:34:8f:b4:cd:9f:ea:b3:d2:42:04:
99:2c:6d:d9:ea:82:4f:bc:a5:91:cd:64:cf:68:a2:
33:ad:05:26:77:5c:98:48:fa:fa:31:52:81:77:e1:
f8:df:91:81:a8:b9:45:08:11:06:fd:58:bd:3d:73:
79:9b:22:95:75:c4:f3:b2:91:01:a0:3e:e1:f0:54:
72:b3:61:57:84:d9:24:4c:e0:ed:63:9c:77:e8:e2:
12:ab:52:ab:dd:f4:a9:28:22:4b:6b:6f:74:b7:11:
47:86:dd:60:71:bd:91:13:d7:87:0c:6b:52:c0:bc:
8b:9c:10:2c:fe:32:1d:ac:35:7e:03:0e:d6:c5:80:
04:0c:a4:1c:13:d6:b4:96:78:11:80:7e:f2:a2:25:
98:3e:a9:f8:8d:67:fa:a4:26:20:f4:2a:4f:5b:db:
e0:3b
prime1:
00:f5:5d:1f:9b:de:51:9d:3d:30:e9:a0:82:0a:08:
50:f3:a6:d0:5e:fa:82:dc:c7:ef:94:8a:09:83:42:
24:ad:6f:b7:7e:e7:92:f5:3b:1e:e6:ef:c0:27:0e:
5e:8f:dd:59:fa:b4:0a:da:83:3d:f0:ea:69:eb:2f:
0b:37:d7:17:a3:fd:03:29:28:0c:d4:75:25:fd:22:
97:0c:8f:d8:60:c7:be:74:94:82:c4:2a:6e:16:a4:
b5:b7:ed:f2:7e:01:21:4f:55:fd:58:39:a1:1c:86:
be:6b:bd:8b:01:e5:dd:ee:f7:43:bc:a5:9e:89:6c:
17:51:14:5d:4b:6a:9d:a9:ed
prime2:
00:f3:4e:8a:f7:a6:b3:b2:d2:90:68:90:db:77:f5:
6b:8c:4c:5a:5a:f4:2b:ec:8c:ac:d7:9f:c4:79:60:
6d:85:ee:b6:41:ec:95:ae:f7:3d:0e:7e:24:f0:da:
91:15:a5:24:d9:78:7d:57:3e:d3:a1:86:8b:ff:27:
65:cc:5b:9e:12:38:a7:93:1b:dd:5a:fd:51:a0:bd:
6b:f9:3f:26:90:9f:ff:6d:93:d3:35:e6:78:59:59:
c1:4f:85:12:05:ad:4a:8c:e9:95:e6:f9:2e:49:0e:
7a:5f:ea:b2:b9:fc:da:00:80:be:dc:17:fa:81:62:
27:f2:d6:03:40:44:c0:c3:63
exponent1:
00:a3:93:6a:67:e9:8b:be:28:cb:46:6b:01:5c:05:
8b:4d:19:e0:3f:51:ac:93:2f:f5:0d:b1:5b:ac:d6:
c3:1e:4a:7a:54:9a:61:f8:d2:14:99:f5:2a:c4:b4:
3f:0a:93:91:51:cd:5c:91:ac:d3:f5:f1:9b:f2:1f:
5c:cf:e4:ba:6d:53:57:70:c5:5d:e2:f8:c3:fe:17:
0f:5d:b5:3a:eb:2f:d4:4d:b8:57:2d:71:9e:b9:c3:
23:cf:f3:f6:fe:ab:6b:8a:39:53:90:26:6b:68:59:
d4:47:d3:b2:01:43:e9:49:fa:2d:28:6e:69:b0:f2:
ba:36:0d:93:87:9c:69:1b:f3
exponent2:
00:a2:34:5c:a5:19:cd:21:e1:b5:9b:0b:3c:fa:a3:
9d:08:32:e6:e7:4d:72:9d:b3:1d:e5:15:2d:a6:40:
49:03:f4:79:81:48:63:c9:fa:28:b4:54:18:a0:91:
b6:0e:6e:18:90:fa:fe:3a:29:e2:6b:af:07:ff:6f:
99:32:e7:be:b6:d0:6f:b7:67:e8:e7:53:8b:c0:7e:
47:fb:7f:6f:0b:15:54:f3:b7:e2:23:ee:fa:e6:3b:
d6:35:03:61:59:1e:31:b3:46:63:ef:50:c9:86:09:
a6:ea:9c:77:26:a8:91:55:ab:29:e8:0f:fc:56:41:
6f:f7:39:57:80:2d:d5:d7:97
coefficient:
00:85:4e:43:ad:27:2d:f5:81:12:c9:84:ca:73:eb:
bf:9a:57:8a:56:53:ff:eb:79:dc:43:a4:20:27:b5:
5a:69:27:d7:74:65:c4:2d:83:e9:d8:ee:9e:7c:bc:
91:6f:ed:76:2a:7b:90:24:a5:bd:ee:8a:f3:1d:38:
d5:c1:84:0d:80:78:13:9a:3c:b0:04:7c:c7:36:cf:
22:4d:99:25:f6:1a:42:fd:0f:48:87:09:e7:24:a5:
c6:f3:21:40:a0:5f:22:e4:9b:37:2d:f7:f4:39:c6:
ad:86:07:89:6b:d6:b3:8a:d5:9e:bd:be:1b:8d:e9:
32:54:35:25:7e:4c:a4:b0:9d
$ openssl rsa -in privkey.pem -noout -text
RSA Private-Key: (2048 bit, 2 primes)
modulus:
00:d3:03:aa:59:c3:24:82:25:f2:ff:52:e3:2e:33:
cd:cb:79:7c:3b:50:1f:ed:5e:51:c0:18:9f:27:ef:
7c:d6:14:c7:c4:cb:f2:07:9e:66:3b:24:f7:ae:2e:
96:83:b4:87:8c:e6:62:d6:c9:44:19:68:74:f4:a4:
84:22:35:33:03:c9:ad:8f:fd:28:24:b4:f7:09:96:
ee:55:8a:89:e1:e9:86:79:30:52:09:a1:24:aa:f6:
2e:ec:d3:18:7d:2d:8e:e2:cf:5e:b1:b5:a1:b7:34:
79:58:aa:53:20:5b:1f:72:68:56:90:3d:6a:95:3e:
d7:2a:74:67:3c:92:f4:6f:52:92:b4:f1:c5:9c:db:
64:5a:6a:fc:4d:03:81:de:c8:04:44:77:56:58:c5:
49:95:bf:0c:ba:58:e5:f9:0f:15:2a:55:71:72:bc:
04:0c:43:2f:e6:3e:de:5b:1c:f1:4f:2f:1a:11:d3:
0d:3a:96:ba:a2:4c:3a:39:82:46:12:92:be:8e:9d:
b3:5f:d0:88:a8:cb:af:0e:26:32:6e:63:0e:e6:90:
36:8c:d0:a8:86:85:02:e9:dd:b1:7d:de:45:17:9c:
e7:bb:22:b1:0f:9d:eb:2e:3d:8d:dc:d3:0c:a9:7c:
9b:ae:84:ca:7c:3c:9a:13:af:a9:5d:e4:78:3e:a2:
08:21
publicExponent: 3 (0x3)
privateExponent:
00:8c:ad:1c:3b:d7:6d:ac:19:4c:aa:37:42:1e:cd:
33:dc:fb:a8:27:8a:bf:f3:94:36:80:10:6a:1a:9f:
a8:8e:b8:85:2d:dd:4c:05:14:44:27:6d:fa:74:1f:
0f:02:78:5a:5d:ee:ec:8f:30:d8:10:f0:4d:f8:6d:
ad:6c:23:77:57:db:c9:0a:a8:c5:6d:cd:fa:06:64:
9e:e3:b1:b1:41:46:59:a6:20:36:b1:16:18:71:f9:
74:9d:e2:10:53:73:b4:97:34:e9:cb:ce:6b:cf:78:
50:e5:c6:e2:15:92:14:f6:f0:39:b5:7e:47:0e:29:
e4:c6:f8:44:d3:0c:a2:f4:e0:80:b9:38:56:88:70:
9e:e8:1d:45:ea:eb:73:b3:f7:c9:56:dd:2b:a0:2e:
0a:9c:24:56:6a:50:02:09:e5:93:a1:ab:11:f3:2d:
b5:a0:66:7b:42:1c:8f:a5:ea:10:86:0b:62:c2:7e:
2e:7f:94:a0:ff:a8:93:fb:4a:23:f2:1f:08:ea:31:
9e:85:c1:0a:06:4a:c8:35:9c:7f:3b:3b:49:d5:34:
be:dc:d3:d4:4d:cf:a4:0e:32:bc:6c:8d:3b:65:4a:
4b:8a:00:c8:a1:0b:56:e4:92:0b:c1:6e:54:41:89:
4b:a1:ea:1c:12:6d:c5:ab:ef:0f:4d:aa:10:d0:2b:
54:63
prime1:
00:f6:45:7c:32:3b:66:f4:3f:a7:a2:0f:03:b0:1a:
8c:09:c2:57:f7:fa:65:c7:ce:6a:01:ac:d8:16:bc:
a4:de:5d:a7:69:2f:d8:2a:c0:36:ca:18:bc:c1:d3:
6e:03:a9:ca:6c:2c:1c:f9:00:7c:10:69:32:c9:46:
cb:51:ac:b1:51:c5:c2:91:23:69:55:3a:70:dd:ea:
74:2b:c5:67:3b:aa:a9:62:f8:02:8c:5e:c7:f4:69:
9b:ff:d6:c7:be:ab:3c:27:84:24:a1:55:48:a4:fc:
e7:59:46:78:40:36:d2:a9:a1:18:69:82:24:b3:f2:
18:e6:6d:98:07:86:25:2c:93
prime2:
00:db:59:a1:11:94:cb:81:be:97:71:5d:9e:a4:36:
48:4c:7f:d3:9c:ee:1a:71:dd:1e:89:6e:08:cc:2d:
91:d9:5a:2d:6d:56:1f:48:db:72:ac:ea:57:45:0e:
cf:d4:dc:53:99:c1:f8:ce:7a:bb:b8:b9:9c:92:f9:
c7:be:7a:b2:df:69:90:b4:73:c5:a4:64:ea:a4:d3:
47:47:d0:22:e4:7c:17:b5:49:90:5d:b3:05:98:6a:
f5:97:04:42:ad:54:71:4e:e7:fc:e2:c8:c4:43:da:
7b:22:f4:35:4e:10:5c:00:9a:8d:36:de:73:3d:9f:
b0:2b:fb:cd:57:80:3b:dc:fb
exponent1:
00:a4:2e:52:cc:27:99:f8:2a:6f:c1:5f:57:ca:bc:
5d:5b:d6:e5:4f:fc:43:da:89:9c:01:1d:e5:64:7d:
c3:3e:e9:1a:46:1f:e5:71:d5:79:dc:10:7d:d6:8c:
f4:02:71:31:9d:72:bd:fb:55:a8:0a:f0:cc:86:2f:
32:36:73:20:e1:2e:81:b6:17:9b:8e:26:f5:e9:46:
f8:1d:2e:44:d2:71:c6:41:fa:ac:5d:94:85:4d:9b:
bd:55:39:da:7f:1c:d2:c5:02:c3:16:38:db:18:a8:
9a:3b:84:50:2a:cf:37:1b:c0:ba:f1:01:6d:cd:4c:
10:99:9e:65:5a:59:6e:1d:b7
exponent2:
00:92:3b:c0:b6:63:32:56:7f:0f:a0:e9:14:6d:79:
85:88:55:37:bd:f4:11:a1:3e:14:5b:9e:b0:88:1e:
61:3b:91:73:9e:39:6a:30:92:4c:73:46:e4:d8:b4:
8a:8d:e8:37:bb:d6:a5:de:fc:7d:25:d1:13:0c:a6:
85:29:a7:21:ea:46:60:78:4d:2e:6d:98:9c:6d:e2:
2f:85:35:6c:98:52:ba:78:db:b5:93:cc:ae:65:9c:
a3:ba:02:d7:1e:38:4b:89:ef:fd:ec:85:d8:2d:3c:
52:17:4d:78:de:b5:92:ab:11:b3:79:e9:a2:29:15:
20:1d:52:88:e5:00:27:e8:a7
coefficient:
00:b1:63:cc:82:98:e6:3e:a2:94:f2:02:88:14:33:
04:ad:c9:a5:55:ca:17:0e:55:94:29:54:77:38:01:
40:56:52:d8:f9:98:da:54:95:41:78:2b:b5:65:ff:
49:b3:2a:e5:f6:d8:14:bf:38:a8:78:c5:98:c6:30:
7f:c2:94:a0:15:1f:26:81:b8:a3:9d:6a:ce:0a:73:
2f:eb:6c:b5:14:a0:3d:e8:d9:22:90:a8:f1:49:92:
31:a9:45:d6:98:c8:46:1c:de:13:15:6d:9b:fc:5b:
09:ca:3b:f0:d2:7a:77:0f:3a:b3:fe:db:37:5b:d4:
13:5c:8d:cb:57:f9:d6:08:81
It can be seen that for the first RSA private key, the numbers dp and c (both styled in bold) are not right. This could be potentially dangerous when signatures are involved.
Because of the Chinese Remainder Theorem, if one uses the incorrectly computed private key to sign a message and sends the resulting wrong signature directly without first verifying it, this can lead to the so-called "fault attack" (e.g., explained here).
Continuing with the example above, if we sign the message Hello World\n using the first RSA private key:
// somehow this prime number leads to incorrect dp, which is quite weird
OCT_fromHex(&P, "f55d1f9bde519d3d30e9a0820a0850f3a6d05efa82dcc7ef948a09834224ad6fb77ee792f53b1ee6efc0270e5e8fdd59fab40ada833df0ea69eb2f0b37d717a3fd0329280cd47525fd22970c8fd860c7be749482c42a6e16a4b5b7edf27e01214f55fd5839a11c86be6bbd8b01e5ddeef743bca59e896c1751145d4b6a9da9ed");
OCT_fromHex(&Q,"f34e8af7a6b3b2d2906890db77f56b8c4c5a5af42bec8cacd79fc479606d85eeb641ec95aef73d0e7e24f0da9115a524d9787d573ed3a1868bff2765cc5b9e1238a7931bdd5afd51a0bd6bf93f26909fff6d93d335e6785959c14f851205ad4a8ce995e6f92e490e7a5feab2b9fcda0080bedc17fa816227f2d6034044c0c363");
// use our choice of P and Q
RSA_2048_KEY_PAIR(NULL,3,&priv,&pub,&P,&Q);
print_keypair(&priv, &pub);
printf("Signing message\n");
OCT_jstring(&M,(char *)"Hello World\n");
PKCS15(HASH_TYPE_RSA_2048,&M,&C);
RSA_2048_DECRYPT(&priv,&C,&S); /* create signature in S */
printf("Signature= ");
OCT_output(&S);
We get a faulty signature: s = 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.
If an attacker obtains this signature, and knows the choice of hash algorithm as well as the message of Hello World\n, then he/she can also compute the hash value and prepare the PKCS1v1.5 structure accordingly: r = 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
And now the attacker can factorize the modulus n by computing GCD(s^e - r, n). In Python3:
import math
# faulty signature
s = 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
# precomputed PKCS1v1.5 structure
r = 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
# modulus
n = 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
# now factorize n
p1 = math.gcd(pow(s,3,n)-r, n)
p2 = n // p1
print("p=",hex(max(p1,p2)))
print("q=",hex(min(p1,p2)))
Which outputs
p= 0xf55d1f9bde519d3d30e9a0820a0850f3a6d05efa82dcc7ef948a09834224ad6fb77ee792f53b1ee6efc0270e5e8fdd59fab40ada833df0ea69eb2f0b37d717a3fd0329280cd47525fd22970c8fd860c7be749482c42a6e16a4b5b7edf27e01214f55fd5839a11c86be6bbd8b01e5ddeef743bca59e896c1751145d4b6a9da9ed q= 0xf34e8af7a6b3b2d2906890db77f56b8c4c5a5af42bec8cacd79fc479606d85eeb641ec95aef73d0e7e24f0da9115a524d9787d573ed3a1868bff2765cc5b9e1238a7931bdd5afd51a0bd6bf93f26909fff6d93d335e6785959c14f851205ad4a8ce995e6f92e490e7a5feab2b9fcda0080bedc17fa816227f2d6034044c0c363
