Cryptography

Key Lengths

RSA

See recommendations from Bruce Schneier in Applied Cryptography (§7.2, [1]). See also [2]

Recommended public-key key lengths (in bits) — Source: Applied Cryptography, 2nd edition

Year	vs. industry	vs. Corporate	vs. Government
1995	768	1280	1536
2000	1024	1280	1536
2005	1280	1536	2048
2010	1280	1536	2048
2015	1536	2048	2048

Generate RSA Keys

Under Linux, Install package racoon. Then you can use plainrsa-gen to generate a RSA key pair:

sudo plainrsa-gen -b 2048 -e 3

# : PUB 0sAQOTQ2zIwqxqjy4LRTwXEHB/WdxMrrcldKBAut3siLnuQMCDFGkwSfOc9v+77ibPDqtJQj0C8nys7+W1gI3o6yht+SjG+m16hZwvwl0Mt81E11Tca6k6py1wNmntxvePtotG3uk6MhqpluJAUeOxIL6YcHLcsgBi19gwHiU1YBFF2Q==
: RSA   {
        # RSA 1024 bits
        # pubkey=0sAQOTQ2zIwqxqjy4LRTwXEHB/WdxMrrcldKBAut3siLnuQMCDFGkwSfOc9v+77ibPDqtJQj0C8nys7+W1gI3o6yht+SjG+m16hZwvwl0Mt81E11Tca6k6py1wNmntxvePtotG3uk6MhqpluJAUeOxIL6YcHLcsgBi19gwHiU1YBFF2Q==
        Modulus: 0x93436cc8c2ac6a8f2e0b453c1710707f59dc4caeb72574a040baddec88b9ee40c08314693049f39cf6ffbbee26cf0eab49423d02f27cacefe5b5808de8eb286df928c6fa6d7a859c2fc25d0cb7cd44d754dc6ba93aa72d703669edc6f78fb68b46dee93a321aa996e24051e3b120be987072dcb20062d7d8301e2535601145d9
        PublicExponent: 0x03
        PrivateExponent: 0x622cf33081c8470a1eb22e280f604aff913d88747a18f86ad5d1e9485b269ed5d5acb84620314d134f5527f419df5f1cdb817e01f6fdc89fee79005e9b4770484de1f0c003dcbeac2290f28f5594022ec0ca86fd0618ec77d0db3f24e0ddd9339a77b1126f3256d9405ce86bcd456f4db2ef0c019a763abee74eb29cb161568b
        Prime1: 0xc37626fcd807b365f62e70d07ad1c2383f0a987f373eca93bbd723bd6676062263fef48a1c99efbb4e2d64d82fecc1756ea3845db786746d9145f5c267931f5d
        Prime2: 0xc0dfb6dd8fa7b43405ba80653c9d7f58f4a208ae7a430028c149eb523fccea9b7b2c6b146eb53795b3879069cd4bd62e7568c651e12b0b4c43e22387ee6c24ad
        Exponent1: 0x824ec4a890052243f9744b35a736817ad4b1baff7a29dc627d3a17d399a40416ed54a306bdbbf527897398901ff32ba39f17ad93cfaef8490b83f92c450cbf93
        Exponent2: 0x80952493b51a7822ae7c5598d313aa3b4dc15b1efc2caac5d631478c2a889c67a772f20d9f237a63cd050af13387e41ef8f08436961cb232d7ec17aff4481873
        Coefficient: 0x80c5560ddad756e413c19fb39c83370dfa3ca5881ebb0b0a5098fbd81b007e20c7b7a104b0aada943d2f5ae64409a9e3b677e10d5c20f414959a621852424b19
  }

Another solution is to use openssl:

openssl genrsa 256 | openssl asn1parse

This gives:

Generating RSA private key, 256 bit long modulus
....+++++++++++++++++++++++++++
..........+++++++++++++++++++++++++++
e is 65537 (0x10001)
    0:d=0  hl=3 l= 171 cons: SEQUENCE          
    3:d=1  hl=2 l=   1 prim: INTEGER           :00
    6:d=1  hl=2 l=  33 prim: INTEGER           :BEF5D2D6550C4CF428A59A9099573D325D350F603E0C538B17CE6AFA7D2513B9
   41:d=1  hl=2 l=   3 prim: INTEGER           :010001
   46:d=1  hl=2 l=  32 prim: INTEGER           :316F66238264EACF126EBCB2CE5F9D41A74C7DE23069107B73CB664B505C0BD9
   80:d=1  hl=2 l=  17 prim: INTEGER           :F83F3AAC16EA5DFDCF8F08C85415EEA3
   99:d=1  hl=2 l=  17 prim: INTEGER           :C4EC941CCE450AF0B6ED7AA8D73225F3
  118:d=1  hl=2 l=  17 prim: INTEGER           :ECCFB472B1B185543809E480E5E5BE2D
  137:d=1  hl=2 l=  17 prim: INTEGER           :8CF9A4AADE8C14E1F0C31FEDA169383B
  156:d=1  hl=2 l=  16 prim: INTEGER           :1E8BD8FD9E1B1FC747B2C269ECA6CAA5

The following script will generate 10 keys for each size in the set {1024 1536 1664 1792 1920 2048 2304 2560 2816 3072 3328 3584 3840 4096}:

#! /bin/bash
#
# Script to generate a batch of RSA keys of various length
#

function gen-one-key()
{
	openssl genrsa $1 | openssl pkcs8 -topk8 -nocrypt -outform DER -out "$2-pk8.der"
	openssl asn1parse -inform DER -in "$2-pk8.der" > "$2-pk8.txt"
	echo -e "\n############### Content of RSA Private Key object ###############\n" >> "$2-pk8.txt"
	openssl pkcs8 -inform DER -in "$2-pk8.der" -nocrypt | openssl asn1parse >> "$2-pk8.txt"
}

for keylength in 1024 1536 1664 1792 1920 2048 2304 2560 2816 3072 3328 3584 3840 4096; do
	for keyidx in $(seq 1 10); do
		keyname="rsakey-${keylength}b-$(printf '%02d' $keyidx)"
		echo "########## gen-one-key $keylength \"$keyname\""
		gen-one-key $keylength "$keyname"
	done
done

Factorize RSA

Using Sage (see [3]):

# RSA 192-bit
mod192=0xbdd0fcbce5f05aae8049f0699443b575c3119a00f712fd67
print "Factoring RSA 192-bit modulus"
print "mod192=",mod192
print "Using factor():"
time mod192.factor()
print "Using ecm.factor():"
time ecm.factor(mod192)

This gives:

  Factoring RSA 192-bit modulus
  mod192= 4654283518078358737104805100407304944292151641869472955751
  Using factor():
  67662411935248621468167032027 * 68786840210963828271650042213
  Time: CPU 12.14 s, Wall: 13.55 s
  Using ecm.factor():
  [67662411935248621468167032027, 68786840210963828271650042213]
  Time: CPU 0.00 s, Wall: 62.04 s

Other method based on the General Number Field Sieve (GNFS). There are several free ports on Linux:

• [4], links to other projects like GGNFS, MSIEVE, YAFU, up-to-date binaries, Python...
• GGNFS, which is based on GMP library.
• kmGNFS - A General Number Field Sieve (GNFS) Implementation, based on NTL library.
• factor-by-gnfs
• YAFU, Yet Another Factorization Utility
• Flint documentation refers to a program mpQS that would implement the quadratic sieve method.