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Biomedical SecurityErwin M. Bakker

Overview

Cryptography: Algorithms

Cryptography: Protocols

Pretty Good Privacy (PGP) / B. Schneier Workshop

Biomedical Security

Biomedical Application Security (guest lecture by Dr. Erik van der Kouwe

Student Presentations

Student Presentations

Grading: Class participation, assignments, technical paper (4 pages), and presentation(s)

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Cryptography: Sharing Secrets

CAESAR a substitution cipher

Secret Key: 3

Plain Text: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Cipher Text: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

https://www.theregister.co.uk/2006/04/19/mafia_don_clueless_crypto/

Mafia boss Bernardo Provenzano's cipher: ‘A’ -> 4, ‘B’ -> 5, etc.

In April 2006, Provenzano was captured in Sicily partly because

messages encrypted using his cipher, were broken.

DWWDFFKL E3(HELP) = KHOSD3(KHOS) = HELP D3 = E26-k

Cryptography: Sharing Secrets

Alice Bob

Eve

C = EK (‘HELLO BOB’)

Secret key K

DK (C) = ‘HELLO BOB’

Secret key K

Crypto-text C

Crypto-Analyst Eve

• Crypto-text only

• Known Plaintext

• Chosen Plaintext

K?

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Enigma

Encryption as a product of permutations:

P the plug-board transformation

U the reflector

L, M, and R the three rotors

Then encryption is E = PRMLUL-1M-1R-1P-1

After each key press the rotors turn i positions changing the

transformation: R becomes CiRC-i, where C is the cyclic permutation

(A->B, B-> C, etc. …)

the military Enigma has

158,962,555,217,826,360,000 settings

http://enigmamuseum.com/replica/https://en.wikipedia.org/wiki/Enigma_machine

ONE-TIME PAD

A crypto system with perfect secrecy

Plaintext: 01000110101110100110

Key: 11010100001100010010

Crypto-text: 10010010100010110100

Uses XOR for both encryption and decryption.

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Classical Symmetric or Two-way Crypto

Systems

A shared secret key K used for both encryption as well as decryption.

Plaintext P

Crypto-text C

C = EK(P)

P = E-1K(C)=DK(C)

Classical Symmetric Crypto System:

Data Encryption Standard (DES)

March 17, 1975 published by the National Bureau of Standards (NBS)

NSA reduced key-size from the original 128-bit to 56-bit

At the time NSA studied it and said it was secure to use as a standard SKCS.

Next government standard was classified: Skipjack

Block cipher encrypting data in 64-bit blocks

Key length 56-bits

16 rounds: in each round a substitution followed by a permutation

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IBM’s Cipher LUCIFER

designed by H. Feistel and

D. Coppersmith in 1973 used

Feistel Networks for encryption

and decryption.

LUCIFER is one of the first

commercial block ciphers

on which DES is based.

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Encryption in DES

12Classical Symmetric Crypto System:

International Data Encryption Algorithm (IDEA)

IDEA is a Block Cipher designed by X. Lai and J. Massey in 1990. Revised in 1991 to withstand differential cryptanalysis.

Block Length

64-bit Data Blocks Is considered safe against statistical attacks. Cipher Feedback Mode enhances cryptographic strength.

128-bit Key

Safe against brute-force attacks.

Good Confusion

By using three operations: XOR, Addition mod 216, Multiplication mod 216+1 (compare with DES: XOR, small S-Boxes)

Good Diffusion

Every plaintext bit and every key-bit influences every ciphertextbit.

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13 Symmetric Cryptosystem: BLOWFISH

Blowfish is a symmetric block cipher

designed by Bruce Schneier in 1993.

Block Length

64-bit data blocks encrypted in 64-bit ciphertext Blocks.

Key Length

32- 448 bits (1 to 14 32-bit key-blocks).

Variable Security

Key generates 18 32-bit subkeys, and 4 8x32 bit S-boxes.

The algorithm itself is used for this.

Fast, simple, and compact

On a 32-bit processor: 18 clock cycles per encrypted bye.

Uses less than 5K of memory (was too big for smart-cards).

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15 Rivest Cipher 5 (RC5)

RC5 is a block-cipher by R. Rivest in 1994.

Efficient Hard and Software Implementations

Simple structure, simple operations, low memory requirements, fast and simple implementations.

Variable Word Length:

w = 16, 32,or 64 Length of the plaintext blocks is 2w

Variable Key-Length

b = 0,..,255 bytes

Variable Security

Depending on the parameters, number of rounds: r = 0,..255

Data-Dependent Rotations

Circular Bit Shifts. RC5-w/r/b = RC5-32/12/16 considered to have “Nominal” Security. Incorporated in the products BSAFE, JSAFE, and S/MAIL of RSA Data Security, Inc.

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Rivest Cipher 5 (RC5) Modes

Block Cipher Mode

Cipher Block Chaining Mode

RC5-CBC-Pad

RC5-CTS Ciphertext Stealing

Mode: CBC style.

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17 CAST-128

A symmetric encryption cipher by

C. Adams and S. Tavares in 1997.

Uses four primitive operations

addition and substraction mod 232, XOR, left circular rotations.

Uses fixed non-linear S-boxes, also for sub-key generation.

A function F is used with good confusion, diffusion, and

avalanche properties.

its strength is based on the S-boxes. F differs per round.

increase of strength of CAST-128 using more rounds is not (yet) demonstrated.

64-bits data blocks

40- 128-bits key

CAST-128 is used in PGP.

18 Rivest Cipher 2 (RC2)

A symmetric encryption cipher by

R. Rivest in 1997.

Designed for 16-bit microprocessors

Uses 6 primitive operations

addition and subtraction mod 232, XOR, COMPL, AND, and Left

Circular Rotation.

No Feistel Structure.

18 rounds: 16 mixing rounds, and 2 mashing rounds.

64-bits data blocks

8 - 1024-bits key

RC2 is used in S/MIME with 40-, 64-, and 128-bits keys.

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19 Characteristics of Advanced Symmetric Block

Ciphers

Variable Key Length Blowfish, RC5, CAST-128, and RC2

Mixed Operators

Data-Dependent Rotation An alternative to S-boxes. No dependence

on sub-keys. RC5.

Key-Dependent Rotation CAST-128

Key-Dependent S-Boxes Blowfish

Lengthy Key Schedule Algorithm Against brute-force attacks. Blowfish

Variable F to complicate cryptanalysis. CAST-128

20 Advanced Symmetric Block Ciphers

Variable Plaintext/Ciphertext Block Length

For convenience and cryptographic strength (longer blocks

is better) RC5

Variable Number of Rounds

More rounds increase cryptographic strength. Trade-off

between execution time and security. RC5

Operation on Both Data Halves in Each Round

AES, IDEA, Blowfish, and RC5

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Advanced Encryption Standard (AES)

Block-size: 128

Key-sizes: 128, 192, 256

NIST Specification 2001

Origin: a subset of 3 out of the Rijndael Cipher by V. Rijmen and J. Daemen

(NIST paper 2003)

Substitution-permutation network

From the cipher key the keys per round are derived.

Each round

has a non-linear substitution step implemented using a lookup table

Followed by transposition using cyclic shifts

And a mixing step on the columns of the internal state matrix.

The round ends with an add key operation.

Public Key Crypto Systems

Idea by Diffie and Hellman [1976]

Encryption method made public.

Decryption method kept secret.

Closely related to the idea of cryptographic one-way functions:

f(x)xeasy

intractable

But there exists a trapdoor that makes it easy to calculate x given f(x).

Note: No known cryptographic one-way functions. Only likely to be intractable!

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Public Key Crypto System: RSA

Key Generation

Select p, q, both primes.

Calculate n = p * q

Calculate φ(n) = (p-1)(q-1)

Select integer e, such that gcd(φ(n) ,e) =1, and 1< e< φ(n)

Calculate d, such that d = e-1 mod φ(n)

Public key is (e, n)

Secret key is (d, n)

Encryption of plaintext M < n

C = Me mod n

Decryption of ciphertext/crypto-text C

M = Cd mod n

Public Key Crypto System: RSA Example

Key Generation

p= 5, q = 11

=> n = p x q = 5x11 = 55 and φ(n) = (p-1)(q-1) = 4x10 = 40

e = 7, is such that gcd(φ(n) ,e) = gcd(40,7) = 1 and clearly 1< 7 < φ(n)

then d = 23 = e-1 mod φ(n) as 7x23 mod 24 = 161 mod 40 ≡ 1

Public key is (7, 55)

Secret key is (23, 55)

Encryption of plaintext M = 3 < n = 55

C = Me mod n = 27 mod 55 = 128 mod 55 = 18 mod 55 ≡ 18

Decryption of ciphertext/crypto-text C

now Cd mod n = 1823 mod 55 = 223x346mod 55 = 4x18x18x18x36mod 55 =

32 x 312mod 55 = 32x263mod 55 = 2x18x133mod 55 = 79092 mod 55 =

1438x55 +2 mod 55 = 2 mod 55 ≡ 2 = M

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RSA Chosen Ciphertext Attack

Note for RSA: Ee(M1)xEe(M2) = Ee(M1xM2)

Assume (e,n) is the public keys and (d,n) the private key.

Assume C ≡ Me mod n is intercepted.

Chosen Ciphertext attack:

Compute X ≡ (C x 2e) mod n

Submit X as a chosen ciphertext and receive back Y ≡ Xd mod n

X ≡ (C x 2e) mod n = (C mod n) x (2e mod n) = (Me mod n) x (2e mod n) = (2M)e mod n

Thus Y ≡ Xd mod n = (2M)ed mod n = 2M mod n

Unsafe Modes of RSA

Unsafe primes, etc.

D. Boneh et al. Why Textbook ElGamal and RSA Encryption Are Insecure, T.

Okamoto (Ed.): ASIACRYPT2000, LNCS 1976, pp. 30–43, 2000:

without proper preprocessing of the plaintexts, both E lGamal and RSA encryption are fundamentally insecure.

when these systems to encrypt a (short) secret key of a symmetric cipher it is often possible to recover the secret key from the ciphertext.

Conclusion: Preprocessing messages prior to encryption is an essential

part of both systems.

Optimal Asymmetric Encryption Padding (OAEP)

Introduced by Bellare and Rogaway in 1994.

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Safe Mode of

RSA

Optimal Asymmetric Encryption Padding (OAEP)

Introduced by Bellare and Rogaway in 1994.

Note:

We can randomly pad the plaintext prior to

encryption,

Then for this adapted version of RSA:

ξe(M1)x ξ e(M2) ≠ ξ e(M1xM2), but this is not enough.

OAEP is a solution:

• P is a set of optional parameters.

• MGF is jus another hash function.

• EM is finally encoded using RSA.

Diffie-Hellman Key Exchange Algorithm Global and Public

Prime q, and α < q a primitive root of q,

i.e., α generates the multiplicative group of integers mod q

Alice Key Generation

Select private 𝑋𝐴 < 𝑞

Calculate public 𝑌𝐴 = α𝑋𝐴𝑚𝑜𝑑 𝑞

Bob Key Generation

Select private 𝑋𝐵 < 𝑞

Calculate public 𝑌𝐵 = α𝑋𝐵𝑚𝑜𝑑 𝑞

Generation of the Exchanged Secret Key by Alice

𝐾 = 𝑌𝐵𝑋𝐴𝑚𝑜𝑑 𝑞

Generation of the Exchanged Secret Key by Bob

𝐾 = 𝑌𝐴𝑋𝐵𝑚𝑜𝑑 𝑞

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Diffie-Hellman Key-Exchange Example α generates the multiplicative group of integers mod q

For example:

If α = 2, q = 7, then α is no generator of ℤ7∗ ={1, 2, 3, 4, 5, 6} as

{20 mod 7, 21 mod 7, 22 mod 7, 23 mod 7, 24 mod 7, 25 mod 7} = {1, 2, 4, 1, 2, 4,1} ≠ ℤ7∗

If α = 3, q = 7, then

{30 mod 7, 31 mod 7, 32 mod 7, 33 mod 7, 34 mod 7, 35 mod} = {1, 3, 2, 6, 4, 5} = ℤ7∗

Note: Assuming the generalized Riemann hypothesis, the least primitive root αp = O(log

6 p) (Shoup,1990, 1992).

Alice Key Generation:

Select private 𝑋𝐴 < 𝑞

Calculate public 𝑌𝐴 = α𝑋𝐴𝑚𝑜𝑑 𝑞

From 𝑌𝐴 it should be difficult to calculate 𝑋𝐴.

Calculating 𝑋𝐴 can be done taking the discrete log of 𝑌𝐴 to the base α modulo q

Diffie-Hellman Key Exchange

Complexity of Discrete Log

Calculating 𝑋𝐴 by taking the discrete log of 𝑌𝐴 to the base α modulo q is assumed to be intractable.

Shor, Peter (1997). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Journal on Computing. 26 (5): 1484–1509.

Adrian, David et al. (October 2015). "Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice“:

Logjam a flaw in TLS that lets man-in-the middle downgrade connections to ‘export-grade’-DH

Precomputations of a week for a 512-bit group => calculate discrete log for that group in ~ 1 minute.

They found that 82% of vulnerable servers use a single 512-bit group, allowing us to compromise connections to 7% of Alexa Top Million HTTPS sites.

WeakDH.org: use a 2048-bit Diffie-Hellman group

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Digital Signatures

Alice Bob

Message M and signature S = DAlice (M)Checks with public EAliceif EAlice(S) = EAlice(DAlice(M)) = M

(M,S)

Using a public key crypto system.

Public Key Crypto System ElGamal

Public Key

p a prime

g < p

𝑦 = 𝑔𝑥𝑚𝑜𝑑 𝑝

Private Key

x < p

Encryption of message M

random k, with gcd(k,p-1)=1

𝐶 = 𝑎, 𝑏 , 𝑤ℎ𝑒𝑟𝑒 𝑎 = 𝑔𝑘𝑚𝑜𝑑 𝑝 a𝑛𝑑 𝑏 = 𝑦𝑘𝑀 𝑚𝑜𝑑 𝑝

Decryption

𝑀 = 𝑏/𝑎𝑥mod p

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ElGamal Signatures

Public Key

p a prime

g < p

𝑦 = 𝑔𝑥𝑚𝑜𝑑 𝑝

Private Key

x < p

Signing of message M

random k, with gcd(k,p-1)=1

Signature S = 𝑎, 𝑏 , 𝑤ℎ𝑒𝑟𝑒

𝑎 = 𝑔𝑘𝑚𝑜𝑑 𝑝 a𝑛𝑑 𝑏 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑀 = (𝑥𝑎 + 𝑘𝑏) 𝑚𝑜𝑑 𝑝-1

Verification

Accept as valid if 𝑦𝑎𝑎𝑏 𝑚𝑜𝑑 𝑝 = 𝑔𝑀𝑚𝑜𝑑 𝑝

34 Cryptographic Hash Functions

An hash function H has the following properties:

H can be applied to data of any size.

H produces fixed length output.

H(x) is easy to compute for any given x.

One-way

for any given hash-code h, it is computationally infeasible to find x such

that H(x) = h.

Weak collision resistance

for any given x, it is computationally infeasible to find y (not equal to x)

such that H(y) = H(x).

Strong collision resistance

it is computationally infeasible to find any pair (x,y) such that H(x) = H(y).

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35 Cryptographic Hash FunctionsAn hash function H has the following properties:

H can be applied to data of any size.

H produces fixed length output.

H(x) is easy to compute for any given x.

One-way

for any given hash-code h, it is computationally infeasible to find x such that H(x) = h.

Weak collision resistance

for any given x, it is computationally infeasible to find y (not equal to x) such that H(y) = H(x).

Strong collision resistance

it is computationally infeasible to find any pair (x,y) such that H(x) = H(y).

Could you use DES to implement a Cryptographic Hash Function?

Secure Hash Algorithm (SHA) Developed by NSA in 1993. Based on MD4.

In 20002 a revised version by NIST. In 2005 SHA-1 started to be phased out by NIST. By

2010 SHA-256, SHA-384, and SHA-512.

Around 2005 an attach were 2 different message could be found using 269 operations

yielding the same SHA-1 hash! (280 operations were expected to be necessary)

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Secure Hash Algorithm

(SHA)

Basic Cryptographic Protocols

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Authentication: Passwords

Password file protected by:

One-way encryption: only encrypted passwords are stored

Access control: password files only accessed only by a few accounts

Passwords cracked by:

Trying default passwords, all short passwords.

Dictionary attacks, license plates, etc.

User’s information: pets, names, children names, etc.

Trojan Horse to bypass access restrictions

Line tap between user and system.

Authentication:

Passwords

Salt:

• No duplicate password visible

• 12-bit salt makes it 4096x more difficult

to guess password

• Prevents a hardware implementation

of crypt

Crypt

• Password -> 56-bit key

• 64-bit block of 0’s as input

• DES modified with the 12-bit salt

• 12 passes output is input for next pass

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Message Authentication Requirements

Disclosure

Release of message contents to any person or process not possessing the appropriate cryptographic key

Traffic analysis

Discovery of the pattern of traffic between parties

Masquerade

Insertion of messages into the network from a fraudulent source

Content modification

Changes to the contents of a message, including insertion, deletion, transposition, and modification

Sequence modification

Any modification to a sequence of messages between parties, including insertion, deletion, and reordering

Timing modification

Delay or replay of messages

Source repudiation

Denial of transmission of message by source

Destination repudiation

Denial of receipt of message by destination

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. From Stallings Cryptography and Network Security 2017

Message Authentication Functions

Two levels of functionality: Hash function

A function that maps a message of any length into a fixed-length hash value which serves as the authenticator

Message encryption

The ciphertext of the entire message serves as its authenticator

Message authentication code (MAC)

A function of the message and a secret key that produces a fixed-length value that serves as the authenticator

Lower level

•There must be some sort of function that produces an authenticator

Higher-level

•Uses the lower-level function as a primitive in an authentication protocol that enables a receiver to verify the authenticity of a message

© 2017 Pearson Education, Inc., Hoboken, NJ. All rights reserved. From Stallings Cryptography and Network Security 2017

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Authentication:

Authenticated Secure Communication

Man in the middle Attack

Or Actually: Eve the Man in the Middle

Alice Bob

Eve

S = Signature(M) = Ed (‘HELLO BOB’)

Message M

Receive M and S

Check that S is from Alice

M, S

…?

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Man-in-the-Middle-Attack-Attack

https://www.bleepingcomputer.com/news/security/google-chrome-will-

soon-warn-you-of-software-that-performs-mitm-attacks/

Number TheoryBasic Definitions and Algorithms

https://www.bleepingcomputer.com/news/security/google-chrome-will-soon-warn-you-of-software-that-performs-mitm-attacks/

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47 Public-Key Cryptography Fast Exponentiation

Calculate ab mod n = 7560 mod 561

a = 7, b = 560 = 1000110000, n = 561

I bi c d ->7560

9 1 1 7 71

8 0 2 49 72

7 0 4 157 74

6 0 8 526 78

5 1 17 160 716+1

4 1 35 241 732+2+1

3 0 70 298 764+4+2

2 0 140 166 7128+8+4

1 0 280 67 7256+16+8

0 0 560 1 7512+32+16

resultExponent

48

Number Theory

Definition (Divisors):

b 0 divides a, if a = mb for some m.

Notation: b|a

Example: divisors of 24 are

1, 2, 3, 4, 6, 8, 12, and 24

The following relations hold:

if a|1, then a = ±1

if a|b and b|a, then a = ±b

any b 0 divides 0

if b|g and b|h, then b|(mg+nh) for arbitrary integers m and n

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49

Number Theory

Definition (Prime Numbers):

An integer p>1 is a prime number if its only divisors are ±1 and ±p.

Theorem: Any positive integer a>1 can be factored in a unique way as:

a = p1a1.p2

a2…ptat,

where p1 > p2 >…> pt are prime,

and ai >0

Example: 91 = 7 x 13,

11011 = 7 x 112 x 13

0 each where,or pP

aaPa p

50

Number Theory

Definition1 (GCD):

The positive integer c is said to be the greatest common divisor of a and b if:

1) c|a and c|b

2) if d|a and d|b, then d|c

Notation: c = gcd(a,b)

Definition2 (GCD):

gcd(a,b) = max[k, such that k|a and k|b]

Example: 192 = 22 x 31 x 42

18 = 21 x 32

gcd(18,192) = 21 x 31 x 40 = 6

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51

Number Theory

Definition1 (Relative Prime):

The integers a and b are said to be relatively prime if gcd(a,b) = 1.

Example:

192 and 18 are not relatively prime:

192 = 22 x 31 x 42

18 = 21 x 32

gcd(18,192) = 21 x 31 x 40 = 6

74 and 75 are relatively prime:

74 = 2 x 37

75 = 3 x 52

gcd(74,75) = 1

52

Number Theory: Modular Arithmetic

Given any positive integer n and any integer a we can write:

a = qn + r, where 0 r < n, q = a/n

r is called the residue (mod n)

Definition: If a is an integer and n is a positive integer we define a mod n to be the remainder when a is divided by n.

Thus, a = a/n x n + (a mod n)

Definition: Two integers are said to be congruent modulo n if

(a mod n) = (b mod n)

Notation: a b mod n

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53

Number Theory: Modular Arithmetic

Examples: 73 4 mod 23 as

73 = 3 x 23 + 4, hence

4 = 73 mod 23, and clearly

4 = 4 mod 23

21 -9 mod 10 as

1 = 21 mod 10 and

1 = -9 mod 10

Properties (Check):

a b mod n if n|(a-b)

(a mod n) = (b mod n) implies a b mod n

a b mod n implies b a mod n

a b mod n and b c mod n implies a c mod n

54

Number Theory: Modular Arithmetic

The mod n operator maps all integers into the set of integers Zn = 0,1,…,(n-1), the set of all residues modulo n.

The following properties hold for modular arithmetic within Zn:

(w +x) mod n = (x + w) mod n

((w+x)+y) mod n = (w+(x+y)) mod n

(0+w) mod n = w mod n

w Zn z Zn such that w + z 0 mod n

(w x) mod n = (x w) mod n

((wx)y) mod n = (w(xy)) mod n

(1 w) mod n = w mod n

(w(x+y)) mod n = ((wx)+(wy)) mod n

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55

Number Theory: Modular Arithmetic

Z8: 0 1 2 3 4 5 6 7

6: 0 6 12 18 24 30 36 42

mod 8: 0 6 4 2 0 6 4 2

Z8: 0 1 2 3 4 5 6 7

5: 0 5 10 15 20 25 30 35

mod 8: 0 5 2 7 4 1 6 3

Note: gcd(6,8) = 2, and gcd(5,8) = 1

Notation: Zp*= 1,2,…,(p-1)

Theorem: Let p prime, then for each w Zp* there exists a

z such that w z 1 mod p,

z is equal to the multiplicative inverse w-1 of w

56

Number Theory: Euler Totient Function

Definition: The Euler’s totient function (n) of n is equal to the

number of positive integers

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57

Number Theory: Euler’s Totient Function

Fermat’s Theorem (1640): For every prime p and any integer a, the following holds:

ap-1 1 mod p.

Euler’s Theorem (~1740): For any positive integer n, and any integer a relative prime to n, the following holds:

a(n) 1 mod n

Corollary: Let p,q be prime, and n = pq, m an integer such that gcd(m,n)=1, then

m(p-1)(q-1) 1 mod n

Examples:

26 = 64 = 63 + 1 1 mod 7

4(5-1)(7-1) = 424 = (48)3mod 35 163mod 35 4096 mod 35 1 mod 35

58

Number Theory: Testing for Primality

[Miller’75, Rabin’80]

Procedure Witness(a,n) n is to be tested for primality, a is some integer less than n.

if (not an-1 1 mod n) or

(x: x2 1 mod n and x1)

then return TRUE {n is no prime}

else return FALSE {n may be prime}

If n is no prime the probability that Witness

returns FALSE is

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59

Number Theory: Number of Primes

Definition: (n) is equal to the number of primes p that satisfy 2pn.

Theorem (The Prime Number Theorem, conjectured by Legendre,

Gauss, Dirichlet, Chebyshev, and Riemann; proven by

Hadamard and de la Vallee Poussin in 1896).

(n)~ n/ln(n)

Thus there are about

10100/ln(10100)-1099/ln(1099) =

0.039x1099 100-digit primes

There are 4.5x1099 100-digit odd numbers.

That is, about 1 of every 115 100-digit odd numbers is prime.

60

Number Theory: Euclid’s Algorithm

Finding the Greatest Common Divisor

Theorem: For any integer a0, and any integer b>0: gcd(a,b) = gcd(b,a mod

b)

Proof: Let d = gcd(a,b) => d|a and d|b => a = kb + a mod b for some integer

k => (a mod b) = a -kb => d|(a mod b) (as d|a and d|kb). Thus d is a

common divisor of b and (a mod b).

Conversely, if d = gcd(b, a mod b), then d|kb and thus also d|(kb + a mod b)

=> d|a. Thus d is also a common divisor of a and b.

qed

Example (Calculation of GCD):

gcd(12,18) = gcd(18,6) = gcd(6,0) = 6

gcd(10,11) = gcd(11,1) = gcd(1,0) = 1

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Number Theory: Euclid’s Extended Algorithm

Finding the Multiplicative Inverse

If gcd(d,n) = 1, then (d-1 mod n) exists.

I.e., dd-1 = 1 mod n.

Complexity: The multiplicative inverse can be found in O(log2n)

time.

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Number Theory: Discrete Logarithm

Definition: Let Zn*={1,2,…,(n-1)}, and g in Zn

*. Then any integer x such that:

gx = y mod n

is called a discrete logarithm of y to base g.

Example:

Z7* 1 2 3 4 5 6

31 32 33 34 35 36

g=3 3 2 6 4 5 1

Z7* 1 2 3 4 5 6

log3 6 2 1 4 5 3

N.B. g=3 is a generator of Z7*

Definition: If for g in Zp* {g1,…,g(p-1)} = Zp

* holds, then g is a generator of Zp*.

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Number Theory: Complexity of PRIMES, Discrete Log, FACTORIZE, etc.

Finding Primes (PRIMES is in P, AKS-Algorithm, August 2002)

After 1/115 tries success. Each try fastexp and some tests are executed => O(log n) time.

Finding Safe Primes

It is unknown whether there exist infinitely many safe primes.

Calculating the Discrete Logarithm

If the prime factors of (p-1) are small there exist efficient algorithms, otherwise roughly the same complexity as factorising.

Factorising n (b-bits)

Peter Shor(1994): O(b3) and O(b) space on a quantum computer.

Kleinjung et al. (2010) used general number field sieve GNFS- approach,

O(𝑒3 64

9𝑏(log 𝑏)2

time, for the factorization of a 768-bit RSA modulus n.

Calculating Euler’s Phi Function of n

It is unknown if this can be done without factorising n.

Finding the multiplicative inverse mod n

O(log2n)

Assignment Set 1

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Set 1 Assignment 1

Note that if the hardware of the Military Enigma would not have been known

you have to take the number of configurations for all rotors and the switch-

board into account.

In practice our key-space size is determined by:

- 5 possible known rotors out of which 3 were selected: 53

= 60 possibilities

- Initial rotor settings (3), (known) reflector was fixed: 263 = 17576

- Notch settings, rotation after each key-stroke: 262 = 676

- Switchboard configuration defined using 10 cables: 150,738,274,937,250

- Key-space size is: 107,458,687,327,250,619,360,000 = 1.07 X 1023

DES Key-space size: 72,057,594,037,927,936 = 7.2x1016

From: http://ciphermachines.com/enigma.html#key

Assignment 2: Differential Cryptanalysis

Idea by Biham and Shamir

A chosen plaintext attack: attacker nows (many) plaintext-cryptotext pairs.

Differential because the difference (through XOR) of 2 plaintexts and their corresponding 2 ciphertexts are taken.

The distribution of these differences give information on key-bits.

After the distributions let you determine several bits with high probability you use brute force on the rest of the key-space ( key-space 247) as the NSA/IBM was aware of this attack in 1974 about 10 years before this was rediscovered.

M.Matsui (1993/1994) Linear Cryptanalysis of DES

8 rounds of DES-algorithm for 211 known plaintexts

16 rounds of DES for 243 known plaintexts

Use this data to find the key (exhaustive search may still be more effective…)

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Set 1 Assignment 3

DES Key-Space size is 256~7.2 𝑥 1016

Titan X performance 11 TFLOPS is 1.1x1013 FLOPS

One 64-bit block DES encryption needs FDES FLOPS

Time to do a known plaintext attack: FDESx7.2 𝑥 1016/ 1.1x1013 = FDESx6.5x10

3sec

… note …

Set 1 Assignment 4

ONE-TIME PAD used twice with same key

P1: 0100011010 P2: 1110100110

Key: 1101010000 Key: 1101010000

C1: 1001001010 C2: 0011110110

C1: 1001001010 P1: 0100011010

C2: 0011110110 P2: 1110100110

C1⨂C2: 1010111100 P1⊗P2: 1010111100

Note: 8-bits often asci-codes