Cryptography: Difference between revisions

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=== Public Domain libraries ===
=== Public Domain libraries ===
* [https://github.com/B-con/crypto-algorithms Crypto-algorithms (github)] — basic implementations from Brad Conte of various algorithms.
* [https://github.com/B-con/crypto-algorithms Crypto-algorithms (github)] — basic implementations from Brad Conte of various algorithms.
* [https://github.com/libtom/libtomcrypt libtomcrypt] ([https://android.googlesource.com/platform/external/dropbear/+/f7fc46c63fdc8f39234fea409b8dbe116d73ebf8/libtomcrypt/src/hashes/sha2/sha256.c also here])
* [https://github.com/libtom/libtomcrypt libtomcrypt] ([https://android.googlesource.com/platform/external/dropbear/+/f7fc46c63fdc8f39234fea409b8dbe116d73ebf8/libtomcrypt/src/hashes/sha2/sha256.c also here]) (tomcrypt)


== Crypto calculators ==
== Crypto calculators ==

Revision as of 13:43, 4 May 2022

References

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

Summary

Src Algorithm Time RAM Code CPU Freq Board Notes
[1] SHA-256 on 1024B 0.6 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] SHA-512 on 1024B 1.4 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-CBC-128 on 1024B 0.7 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-CBC-192 on 1024B 0.8 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-CBC-256 on 1024B 0.9 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-GCM-128 on 1024B 1.8 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-GCM-192 on 1024B 1.9 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-GCM-256 on 1024B 2.0 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-CCM-128 on 1024B 1.7 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-CCM-192 on 1024B 1.9 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] AES-CCM-256 on 1024B 2.1 ms ARM Cortex-M3 96 MHz NXP LPC1768
[1] NIST P-192 ECDHE 229.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-224 ECDHE 303.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-256 ECDHE 432.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-192 ECDSA Verify 251.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-192 ECDSA Sign 66.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-224 ECDSA Verify 329.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-256 ECDSA Verify 458.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-256 ECDSA Sign 122.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-384 ECDSA Verify 806.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-521 ECDSA Verify 1414.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Optim: window=7
[1] NIST P-192 ECDHE 796.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-224 ECDHE 1119.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-256 ECDHE 1672.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-384 ECDHE 3254.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-512 ECDHE 6537.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-192 ECDSA Sign 225.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-224 ECDSA Sign 224.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-256 ECDSA Sign 459.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-384 ECDSA Sign 384.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-512 ECDSA Sign 521.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-192 ECDSA Verify 845.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-224 ECDSA Verify 1185.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-256 ECDSA Verify 1759.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-384 ECDSA Verify 3361.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-512 ECDSA Verify 6693.0 ms ARM Cortex-M0 48 MHz ST Nucleo F091 Optim: NIST, fixed point, window=7
[1] NIST P-192 ECDHE 1155.0 ms ARM Cortex-M3 32 MHz ST Nucleo L152RE Optim: window=7
[1] NIST P-224 ECDHE 1609.0 ms ARM Cortex-M3 32 MHz ST Nucleo L152RE Optim: window=7
[1] NIST P-256 ECDHE 2399.0 ms ARM Cortex-M3 32 MHz ST Nucleo L152RE Optim: window=7
[2] NIST P-192 ECDH 438.9 ms ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 No optim
[2] NIST P-192 ECDH 175.7 ms 2170 B ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 Square + ASM optim
[2] NIST P-256 ECDH 465.1 ms 2512 B ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 Square + ASM optim
[2] NIST P-384 ECDH 1370.3 ms 2244 B ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 Square + ASM optim
[2] NIST P-192 ECDSA Verify 217.1 ms 3014 B ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 Square + ASM optim
[2] NIST P-256 ECDSA Verify 555.2 ms 3334 B ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 Square + ASM optim
[2] NIST P-384 ECDSA Verify 1576.1 ms 3158 B ARM Cortex-M0 32cy mult 48 MHz NXP LPC1114 Square + ASM optim
[1] NIST P-256 ECDSA Sign 122.0 ms 4568 B ARM Cortex-M3 96 MHz NXP LPC1768 Optim: NIST, FP, W=6
[1] NIST P-256 ECDSA Sign 378.0 ms 2972 B ARM Cortex-M3 96 MHz NXP LPC1768 Optim: NIST, W=2
[1] NIST P-256 ECDSA Verify 458.0 ms 5380 B ARM Cortex-M3 96 MHz NXP LPC1768 Optim: NIST, FP, W=6
[1] NIST P-256 ECDSA Verify 759.0 ms 3072 B ARM Cortex-M3 96 MHz NXP LPC1768 Optim: NIST, W=2
[1] NIST P-256 ECDHE 431.0 ms 5012 B ARM Cortex-M3 96 MHz NXP LPC1768 Optim: NIST, FP, W=6
[1] Curve15519 ECDHE 552.0 ms ARM Cortex-M0+ 48 MHz FRDM-KL46Z Google-donna impl
[1] Curve15519 ECDHE 94.0 ms ARM Cortex-M3 96 MHz NXP LPC1768 Google-donna impl
[1] Curve15519 ECDHE 58.0 ms ARM Cortex-M4 120 MHz FRDM-K64F Google-donna impl

Sources:

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
More bench
(see SharkSSL_Benchmark_for_ARM_Cortex_M3.pdf)
(Performance for AES, ECC, ECDH, ECDSA, EdDSA on Cortex M0/M3 ...)


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

libgcrypt

libgcrypt is a very nice C crypto library. It is used in gnupg.

  • Big numbers.
  • symmetric crypto.
  • Asymmetric crypto (RSA, ECC).
  • Fast or secure implementation

Resources:

Reading these tests is the best way to understand how to use the library.

NTRU

BouncyCastle

  • Free Java crypto library

Crypto++

  • A crypto library in C++

Public Domain libraries

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) ))

AES

AES-GCM

Nicer illustration.
A clean C implementation.
In particular describes the "Forbidden Attack" that leads to retrieving H when repeating the IV.
Describes two forgeries attacks and how to retrieve H, ie. the zero block encrypted with the key. Retrieving H means total loss of authentication assurance.

RSA

References

  • ✔Dan Boneh, Twenty Years of Attacks on the RSA Cryptosystem, February 1999, Notices of the AMS, http://www.ams.org/notices/199902/boneh.pdf
  • ✔Don Coppersmith, Small solutions to polynomial equations, and low exponent rsa vulnerabilities, Journal of Cryptology 10 (1997), no. 4, 233–260. 1997-JOC-Vol10-4-002.pdf
    This paper covers the 2 papers:
    • Don Coppersmith, Finding a Small Root of a Bivariate Integer Equation; Factoring with High Bits Known, U. Maurer (Ed.): Advances in Cryptology - EUROCRYPT '96, LNCS 1070, pp. 178-189, 1996. 1996-EUROCRYPT-016.pdf
    • Don Coppersmith, Finding a Small Root of a Univariate Modular Equation, U. Maurer (Ed.): Advances in Cryptology - EUROCRYPT ’96, LNCS 1070, pp. 155-165, 1996. 1996-EUROCRYPT-014.pdf
  • ? Seong-Min Hong, Sang-Yeop Oh, and Hyunsoo Yoon, New Modular Multiplication Algorithms for Fast Modular Exponentiation, Proceeding EuroCrypt'96 (U. Maurer,ed.), Lecture Notes in Computer Science, vol. 1070, Springer-Verlag, 1996, pp. 166--177.
  • ~ Alexander May, New RSA Vulnerabilities Using Lattice Reduction Methods, PhD thesis, University of Paderborn, October 2003. http://www.cs.uni-paderborn.de/uploads/tx_sibibtex/bp.pdf
(Coppersmith method, breaking RSA from partial knowledge of message, partial export of private exponent or prime factors, variants of RSA...)
  • ~ Leo Reyzin, Notes for lecture 8 — Chinese Remainder Theorem and Blum-Blum-Shub PRG, Fall 2004, BU CAS CS 538, http://www.cs.bu.edu/~reyzin/teaching/f04cs538/notes8.pdf
  • ~ Scott A. Vanstone and Robert J. Zuccherato, Short RSA Keys and Their Generation, J. Cryptology 8 (1995), no. 2, 101--114. 1995-JOC-Vol08-2-003.pdf
(Generate modulus with given modulus bit pattern, but the methods in this paper are inefficient and broken - better alternatives below)
  • ? RSA Moduli with a Predetermined Portion: Techniques and Applications, Information Security Practice and Experience (ISPEC 2008), vol. 4991 of Lecture Notes in Computer Science, pp. 116{130, Springer, 2008. http://joye.site88.net/papers/Joy08rsacompr.pdf
  • ~ Arjen K. Lenstra. Generating RSA moduli with a predetermined portion. In Advances in Cryptology - ASIACRYPT'98, volume 1514 of Lecture Notes in Computer Science, pages 1{10. Springer, 1998.
(Implement obvious method; pick n' with the fixed pattern. pick p random prime, round n' up to nearest multiple, compute q' s.t. n'=pq', find nearest prime q, compute n=pq)
Publications by authors

Generate RSA Keys

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

sudo plainrsa-gen -b 512 -e 65537

$ plainrsa-gen -b 512 -e 65537
: RSA	{
	# RSA 256 bits
	# pubkey=0sAwEAAZI52+jaMTOU7BVFJfR3XO0/HNuagkdwnODaOEz5Vl57
	Modulus: 0x9239dbe8da313394ec154525f4775ced3f1cdb9a8247709ce0da384cf9565e7b
	PublicExponent: 0x010001
	PrivateExponent: 0x6a32c54916f676dce89d060c6bc128e6384ddd3480ebc38abb26b06bbbc39ee9
	Prime1: 0xc2ebb37492f49c2536d1425a1e98bced
	Prime2: 0xc00bedf79253f91bc219fb076fdb8407
	Exponent1: 0x6a70e3d268dd82d71f942e33a039b011
	Exponent2: 0x143e2dab36e55b10adf90718d59591e9
	Coefficient: 0x3d988139c172a1b329850a294347b99e
  }

Another solution is to use openssl: 

openssl genrsa 256 > private.pem

Generating RSA private key, 256 bit long modulus
......+++++++++++++++++++++++++++
.............+++++++++++++++++++++++++++
e is 65537 (0x10001)

This produce a private key in .PEM format 

cat private.pem

-----BEGIN RSA PRIVATE KEY-----
MIGqAgEAAiEArBZ8o1wXjfGaJoYyEw20HewqLGTJ6mI3i2ntexqztc8CAwEAAQIg
ObfNFAmGSPB40GUAFI3rE/VQokHFKOpqBnu2/fEGwdECEQDX5IwHMajhOd/e4h1O
J//ZAhEAzA6pCZiM+kcFPDo/S/wB5wIQAlGyL2GZLtIwVXSYW/6SAQIQZQ/VtETz
fXjzJNMMSkuzfQIRAM0VX/ffPwICmDEJGxg5YqY=
-----END RSA PRIVATE KEY-----

See below for how to display components or RSA keys.

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

View and convert RSA Keys

Use openssl rsa -text to display the key 

openssl rsa -text <private.pem        # or openssl rsa -in private.pem -text

Private-Key: (256 bit)
modulus:
    00:ac:16:7c:a3:5c:17:8d:f1:9a:26:86:32:13:0d:
    b4:1d:ec:2a:2c:64:c9:ea:62:37:8b:69:ed:7b:1a:
    b3:b5:cf
publicExponent: 65537 (0x10001)
privateExponent:
    39:b7:cd:14:09:86:48:f0:78:d0:65:00:14:8d:eb:
    13:f5:50:a2:41:c5:28:ea:6a:06:7b:b6:fd:f1:06:
    c1:d1
prime1:
    00:d7:e4:8c:07:31:a8:e1:39:df:de:e2:1d:4e:27:
    ff:d9
prime2:
    00:cc:0e:a9:09:98:8c:fa:47:05:3c:3a:3f:4b:fc:
    01:e7
exponent1:
    02:51:b2:2f:61:99:2e:d2:30:55:74:98:5b:fe:92:
    01
exponent2:
    65:0f:d5:b4:44:f3:7d:78:f3:24:d3:0c:4a:4b:b3:
    7d
coefficient:
    00:cd:15:5f:f7:df:3f:02:02:98:31:09:1b:18:39:
    62:a6
writing RSA key
-----BEGIN RSA PRIVATE KEY-----
MIGqAgEAAiEArBZ8o1wXjfGaJoYyEw20HewqLGTJ6mI3i2ntexqztc8CAwEAAQIg
ObfNFAmGSPB40GUAFI3rE/VQokHFKOpqBnu2/fEGwdECEQDX5IwHMajhOd/e4h1O
J//ZAhEAzA6pCZiM+kcFPDo/S/wB5wIQAlGyL2GZLtIwVXSYW/6SAQIQZQ/VtETz
fXjzJNMMSkuzfQIRAM0VX/ffPwICmDEJGxg5YqY=
-----END RSA PRIVATE KEY-----

... or openssl asn1parse to display the key 

openssl asn1parse <private.pem

    0:d=0  hl=3 l= 170 cons: SEQUENCE          
    3:d=1  hl=2 l=   1 prim: INTEGER           :00
    6:d=1  hl=2 l=  33 prim: INTEGER           :AC167CA35C178DF19A268632130DB41DEC2A2C64C9EA62378B69ED7B1AB3B5CF
   41:d=1  hl=2 l=   3 prim: INTEGER           :010001
   46:d=1  hl=2 l=  32 prim: INTEGER           :39B7CD14098648F078D06500148DEB13F550A241C528EA6A067BB6FDF106C1D1
   80:d=1  hl=2 l=  17 prim: INTEGER           :D7E48C0731A8E139DFDEE21D4E27FFD9
   99:d=1  hl=2 l=  17 prim: INTEGER           :CC0EA909988CFA47053C3A3F4BFC01E7
  118:d=1  hl=2 l=  16 prim: INTEGER           :0251B22F61992ED2305574985BFE9201
  136:d=1  hl=2 l=  16 prim: INTEGER           :650FD5B444F37D78F324D30C4A4BB37D
  154:d=1  hl=2 l=  17 prim: INTEGER           :CD155FF7DF3F02029831091B183962A6

Use option -pubout to extract public keys from private key (PEM format): 

openssl rsa -in private.pem -pubout -out public.pem

writing RSA key

To view a public key from a PEM file, use options -pubin -text 

openssl rsa -in public.pem -pubin -text

Public-Key: (256 bit)
Modulus:
    00:ac:16:7c:a3:5c:17:8d:f1:9a:26:86:32:13:0d:
    b4:1d:ec:2a:2c:64:c9:ea:62:37:8b:69:ed:7b:1a:
    b3:b5:cf
Exponent: 65537 (0x10001)
writing RSA key
-----BEGIN PUBLIC KEY-----
MDwwDQYJKoZIhvcNAQEBBQADKwAwKAIhAKwWfKNcF43xmiaGMhMNtB3sKixkyepi
N4tp7Xsas7XPAgMBAAE=
-----END PUBLIC KEY-----

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

Elliptic Curve Cryptography (ECC)

Standards:

Specifies the recommended curves for ECC (NIST Curve P-256, Curve P-384...).
About deterministic (EC)DSA, including nice test vectors.

Very nice introduction by Andrea Corbellini:

(including nice graphics and visuals with HTML5/Javascript, source available)

ECC Security
References
Curves
  • Weierstrass curves — y^2 = x^3 + a*x + b, the most general form.
  • Montgomery curves — B*y^2 = x^3 + A*x^2 + x, a subset of curves (~40%).
  • Twisted Edwards curves — a*x^2 + y^2 = 1 + d*x^4*y^4, a subset of curves (~40%).
Point coordinates
  • Affine coordinates — The standard (x,y) coordinates used in curve's equation. These coordinates must be completed with the point-at-infinity, which has no affine representation.
  • Projective coordinates — Triplet (x,y,z). Given affine coordinate (X,Y), we have X=x/z; Y=y/z. Point at infinity has coordinate (0,y,0), ie z=0. Using projective coordinates same modular inverse operation (dividing by z).
Implementations
  • A comb method to render ECC resistant against Side Channel Attack, shttps://eprint.iacr.org/2004/342.pdf

Curve25519, edwards25519, Ed25519, X25519

  • X25519 is ECDH on Curve25519.
  • Curve25519, see RFC7748.
  • On existence of equivalent key, see this StackExchange question. We must find a point Q whose order divides 8, then P and P+Q are public keys that will produce the same shared secret for any private scalar k (clamped scalar as in RFC, ie. multiple of 8).
  • Ed25515 is EdDSA on Edwards25519, a variant of Curve25519 (ie. same curve but with transformed coordinates or something).
  • Curve25519 is a Montgomery curve, and edwards25519 is another one that is birational equivalent.
  • X25519 is ECDH built on top of Curve2559. The base point P uses u=9 (v is not used).
  • Ed25519 is EdDSA built on top of edwards25519. The base point Q matches P, and we have y=4/5.

Quantum Cryptography

Quantum Verification Problem
Provably random numbers
Classical Homomorphic Encryption for Quantum Circuits

Post-Quantum Cryptography

Stateful Hash-based signatures
  • NIST SP 800-208 Recomendation for Stateful Hash-Bas Signature Schemes
  • XMS and HMSS? RFC?
  • Security well understood (against quantum attacks).
  • (Very) long-life signatures, but only cover signatures, and limited number of signatures.
  • Suitable to eg. secure firmware update.
Code-based cryptography
  • Based on error-correcting codes (proposed by McEliece in 1978)
  • Quite fast, but very large key sizes (megabytes).
  • NIST candidates: Classic McEliece (KEM), NKE, HQC (alternate KEM).
Lattice-based cryptography
  • Most finalists in NIST PCQ are LBC.
  • Very strong security proofs based on worst-case hardness.
  • Shortest Vector Problem.
  • Find the shortest non-zero vector in an n-dim lattice.
  • Subject to LLL algorithm, but doesn't apply usually (eg. NTRU immune).
  • No known quantum algorithms in poly time.
  • Two basic hard problems:
  • Module-Learning-With-Errors (Module-LWE): problem of finding private key s, given several pairs (a, <a.s> + e), with e error vector.
  • Module-Learning-with-Rounding (Module-LWR).

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

Embedded:

  • RSA-KEM-2048 bit: 28us.
  • RSA-CERT:

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.

Fast hash (non-cryptographic)

See also HN comments on more functions

Custom Cryptographic Algorithms

This section is about variants of standard algorithms.

AES with secret S-boxes

Using AES with secret S-boxes may be interesting to protect against side-channel attacks as long as the S-boxes remain secret (and that adequate sharing on the linear part is in place).

References:

Attack relevant for custom AES. Also state-of-the-art review on custom AES cryptanalysis.

Random Number Generator (rng)

More references:

Linear Congruential Generator

Linear Congruential Generator (wikipedia) are easy to implement and provide good overall properties given the very low cost.

One the most known LCR are the one defined by Knuth 

// https://en.wikipedia.org/wiki/Linear_congruential_generator
// https://stackoverflow.com/questions/23317439/integration-knuth-random-number-generator-to-my-code
// n = (n * 6364136223846793005 + 1442695040888963407) & 0xFFFFFFFFFFFFFFFF;
#define MY_RAND_MAX 0xffffffffffffffffLL

static
unsigned long long _myRandseed = 1;

// DO NOT TRUNCATE!
static
unsigned long long int (myrand)(void)
{
    _myRandseed = _myRandseed * 6364136223846793005 + 1442695040888963407;
    return((unsigned long long int) _myRandseed & MY_RAND_MAX);
}
 ✐  To truncate a LCG RNG, one must take the most significant bits, not the least significant.Linear Congruential Generator (wikipedia) Otherwise the RNG period is considerably reduced.

Linear Feedback Shift Register

Another easy way to produce RNG

Use simple hash function

In spinning-balls, one of Chrome’s benchmarks, we can see an example of this [8]: 

// v8/benchmarks/spinning-balls/v.js

// To make the benchmark results predictable, we replace Math.random
// with a 100% deterministic alternative.
Math.random = (function () {
  var seed = 49734321
  return function () {
    // Robert Jenkins' 32 bit integer hash function.
    seed = seed & 0xffffffff
    seed = (seed + 0x7ed55d16 + (seed << 12)) & 0xffffffff
    seed = (seed ^ 0xc761c23c ^ (seed >>> 19)) & 0xffffffff
    seed = (seed + 0x165667b1 + (seed << 5)) & 0xffffffff
    seed = ((seed + 0xd3a2646c) ^ (seed << 9)) & 0xffffffff
    seed = (seed + 0xfd7046c5 + (seed << 3)) & 0xffffffff
    seed = (seed ^ 0xb55a4f09 ^ (seed >>> 16)) & 0xffffffff
    return (seed & 0xfffffff) / 0x10000000
  }
})()

Mersenne Twister

See https://en.wikipedia.org/wiki/Mersenne_Twister.

Combined

The idea is to take two or more simple RNG, and combine them (eg. by exclusive-OR).

PCG

Reference: PCG, A Family of Better Random Number Generators 

// *Really* minimal PCG32 code / (c) 2014 M.E. O'Neill / pcg-random.org
// Licensed under Apache License 2.0 (NO WARRANTY, etc. see website)
// https://www.pcg-random.org/download.html

typedef struct { uint64_t state;  uint64_t inc; } pcg32_random_t;

uint32_t pcg32_random_r(pcg32_random_t* rng)
{
    uint64_t oldstate = rng->state;
    // Advance internal state
    rng->state = oldstate * 6364136223846793005ULL + (rng->inc|1);
    // Calculate output function (XSH RR), uses old state for max ILP
    uint32_t xorshifted = ((oldstate >> 18u) ^ oldstate) >> 27u;
    uint32_t rot = oldstate >> 59u;
    return (xorshifted >> rot) | (xorshifted << ((-rot) & 31));
}

Quasi-random sequence

An impressive study on how to generate random-looking sequence with better sampling properties than uniform distribution while less regular than lattice.

Useful algorithms

Fisher–Yates shuffle - Random permutation of a finite sequence

-- To shuffle an array a of n elements (indices 0..n-1):
for i from n−1 downto 1 do
     j ← random integer such that 0 ≤ j ≤ i
     exchange a[j] and a[i]

Bloom filters

Bloom filters are very compact and efficient data structure that can tell whether an element is in a set with definitive no answer, or probable yes (with some predefined probability).

Attacks

Rainbow tables

  • See [11], [12].
  • Goal is to reduce the cost of storing big pwd-hash tables.
  • How:
  • Use the hash function h=H(p), and a reduction function p=R(h).
  • Build a chain p0 -> ... -> h_{n-1}, with intermediate p_i, h_i.
  • To avoid loop, the reduction function at each step is different. So even if two hashes reduces to same pwd (and hence to same hash), next reduction will likely generate a different pwd.
  • Build several chains of fixed length.
  • Start searching in the chain backward.
  • Best fix: use salt.
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