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=== Factorize RSA ===
=== Factor RSA modulus ===
Using '''[[Sage]]''' (see [http://sci.tech-archive.net/Archive/sci.math.symbolic/2008-03/msg00023.html]):
Using '''[[Sage]]''' (see [http://sci.tech-archive.net/Archive/sci.math.symbolic/2008-03/msg00023.html]):
<source lang="text">
<source lang="text">

Revision as of 09:53, 30 January 2014

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

Crypto performance

SW Speed on ARM

Hash Algorithm
Name Throughput (kB/s)
SHA-1 1915
MD5 3516
Symmetric algorithms
Name Throughput (kB/s)
AES-CBC 825
AES-ECB 874
AES-CCM, CT only 373
AES-CCM, AD only 816
3DES-CBC 326
3DES-CTR 317
3DES-ECB 333
Asymmetric, Encrypt/Decrypt
Name Time (s)
RSA-1024 encrypt 0.01
RSA-1024 decrypt 0.27
RSA-2048 encrypt 0.05
RSA-2048 decrypt 2.13
Asymmetric, Sign/Verify
Name Time (s)
RSA-1024 sign 0.27
RSA-1024 verify 0.01
RSA-2048 sign 2.13
RSA-2048 verify 0.05
DSA-1024 sign 0.17
DSA-1024 verify 0.33
Diffie-Hellman
Modulus Size (b) Private Key size (b) Time (s)
1024 160 0.17
1024 1024 1.08
2048 224 0.93
2048 2048 8.48

HW Speed on NXP SMX P5Cx08x

Based on linecard figures:

Algo Size Sign Verify
RSA 1024-bit 99 ms (CRT) 2 ms
ECC 192-bit 20 ms 30 ms
DES3 <40 µs <40 µs
AES 128/192/256 12/13/15 µs 12/13/15 µs

Crypto Libraries

NTRU

BouncyCastle

  • Free Java crypto library

Crypto++

  • A crypto library in C++

Crypto calculators

Online

OpenSSL

#Computing AES-128 CBC No padding
echo "000102030405060708090a0b0c0d0e0f000102030405060708090a0b0c0d0e0f" | xxd -r -p |openssl aes-128-cbc -iv 0 -K 01010101020202020303030304040404 -nopad | xxd -p

#Advanced mumbo-jambo
echo $(( (0x$(echo "1111111111f2222222222f3333333333f4444444444f5555555555f6666666666f7777777777f8888888888f9999999999f0000000000f80"|xxd -r -p |openssl des-cbc -iv 0 -K 0102030405060708 -nopad |xxd -p|tail -c 6) & 0x03ffff) + (0x10*2**18) ))

RSA

References

(Coppersmith method, breaking RSA from partial knowledge of message, partial export of private exponent or prime factors, variants of RSA...)

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

Factor RSA modulus

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:

For instance, using YAFU. First factorization is for RSA 192-bit, second is for RSA 256-bit: 

$ unzip yafu-1.19.2.zip
$ cd yafu-1.19.2
$ chmod +X yafu-*
$ ./yafu-64k-linux64

>> factor(4654283518078358737104805100407304944292151641869472955751)

factoring 4654283518078358737104805100407304944292151641869472955751

...

Total factoring time = 8.2085 seconds

***factors found***

PRP29 = 67662411935248621468167032027
PRP29 = 68786840210963828271650042213

or even better with multi-threading: 

$ echo "factor(67838243504816110168272546172330833508240822615334162379358840774428225237019)" | ./yafu-64k-linux64 -threads 24

factoring 67838243504816110168272546172330833508240822615334162379358840774428225237019

...

Total factoring time = 17.05 seconds

***factors found***

PRP39 = 255668459558430779725491264793137830843
PRP39 = 265336770996237319406691555335704774433

Polynomials Equations over F2

Misc Speed Info

HW LM
M/s		 NTLM
M/s		 MD5
M/s		 Ref
NVidia Tesla S1070 680 2600 1920 [5]
NVidia GTX 295 250 1330 880-920 [6]
NVidia GTX 285 195 795 570-585 [7]
Intel Q6600 32 87 70

Ciphers

Bilateral ciphers

Example: http://www.cabinetmagazine.org/issues/40/sherman.php

In this example, people on a photograph are forming a coded phrase by facing forward or sideways, using the code:

code meaning code meaning code meaning code meaning
aaaaa A aaaab B aaaba C aaabb D
aabaa E aabab F aabba G aabbb H
abaaa I/J abaab K ababa L ababb M
abbaa N abbab O abbba P abbbb Q
baaaa R baaab S baaba T baabb U/V
babaa W babab X babba Y babbb Z
Sir Francis Bacon Bilateral code

This code was invented by Sir Francis Bacon. The power of that code is that a's and b's in a message can easily be hidden: he allowed the a’s and b’s in his system to designate the different forms of anything that can be divided into two classes, sorts, or types (which Bacon referred to as the a-form and the b-form). Examples of a/b-forms are: colors of flower, size of objects,


Stream Cipher

Security Properties

  • Stream cipher building block must be invertible, otherwise it is easy to create collisions.


Hash Functions

Security Attacks

  • Man-in-the-Middle pre-image attacks.
Principle is to generate a message m = m1||m2, such that H(m)=h. If H(m)=g(F(IV,m1),m2), the MITM attacks consists in generate random m1, m2 until one get G-1(h,m2) = F(IV,m1). Power of the attack relies on the fact that probability of finding a collision is inv. prop. to sqrt of the state size.
'Countermeasures' - prevent attacker to exploit symmmetry properties between round so that he can't discard part of the state, or control part of the state. Make attacker to use too much memory.
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