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Number Theory and Advanced Cryptography 1. Finite Fields and AES

Chih-Hung Wang

Sept. 2011

Part I: Introduction to Number TheoryPart II: Advanced Cryptography

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Group A set of elements or “numbers” obeys:

(A1) Closure: If a and b belong to G, then ab is also in G.

(A2) Associative: (ab) c = a(b c) (A3) Identity element: There is an element e in G such

that a e = e a = a (A4) Inverses element: For each a in G there is an

element a’ in G such that a a’ = a’ a = e If commutative (A5) a b = b a for all a, b in G then

forms an abelian group

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Cyclic Group Define exponentiation as repeated application of

operator example: a-3 = a a a

Define identity: e=a0

a-n=(a’)n

A group is cyclic if every element is a power of some fixed element ie b = ak for some a and every b in group G

a is said to generate the group G or to be a generator of G.

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Ring A set of “numbers” with two operations (addition + and multiplication

) which are: An abelian group with addition operation (A1-A5) Multiplication:

(M1) Closure (M2) Associative: a(bc)=(ab)c (M3) Distributive law: a(b+c) = ab + ac

If multiplication operation is commutative, it forms a commutative ring (M4) Commutativity of multiplication: ab=ba

If multiplication operation has identity and no zero divisors, it forms an integral domain (M5) Multiplicative identity: There is an element 1 in R such that

a1=1a =a (M6) No zero divisors: If a,b in R and ab=0, then either a=0 or b=0.

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Field A set of numbers with two operations:

Abelian group for addition (A1-A5) Abelian group for multiplication (ignoring 0) (M1-

M6) (M7) Multiplicative inverse: For each a in F,

except 0, there is an element a-1 in F such that

aa-1=(a-1)a =1.

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Group, Ring and Field

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Modular Arithmetic Define modulo operator a mod n to be remainder

when a is divided by n Use the term congruence for: a ≡ b mod n

when divided by n, a & b have the same remainder eg. 73 ≡ 4 mod 23

r is called the residue of a mod n since with integers can always write: a = qn + r

Usually have 0 <= b <= n-1 -12 mod 7 ≡ -5 mod 7 ≡ 2 mod 7 ≡ 9 mod 7

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The Relationship

a = qn + r, 0r<n

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Modulo 7 Example... -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ...

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Divisors Say a non-zero number b divides a if for

some m have a=mb (a,b,m are all integers) That is b divides into a with no remainder Denote this b|a Also say that b is a divisor of a eg. all of 1,2,3,4,6,8,12,24 divide 24

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Modular Arithmetic Operations is 'clock arithmetic' uses a finite number of values, and loops back

from either end modular arithmetic is when do addition &

multiplication and modulo reduce answer can do reduction at any point, ie

a+b mod n = [(a mod n) + (b mod n)] mod n a-b mod n = [(a mod n) – (b mod n)] mod n ab mod n = [(a mod n) (b mod n)] mod n

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Property

ncancbnba

nabnba

bannba

mod imply mod and mod

mod implies mod

)(| if mod

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Modular Arithmetic Can do modular arithmetic with any group of

integers: Zn = {0, 1, … , n-1} form a commutative ring for addition with a multiplicative identity note some peculiarities

if (a+b)≡(a+c) mod n then b≡c mod n but (ab)≡(ac) mod n then b≡c mod n only if a is

relatively prime to n

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Relatively Prime Relative prime: their only common positive integer

factor is 1. An integer has a multiplicative inverse in Zn if that

integer is relatively prime to n.

Example: 63=18 ≡ 2 mod 8 67=42 ≡ 2 mod 8 3 ≡ 7 mod 8

ncb

nacaaba

mod

mod ))(())(( 11

6 and 8 are not relatively prime

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Residue Class The residue classes modulo n as

[0], [1], [2], …, [n-1] where [r] = {a: a is an integer, a ≡ r mod n}

Z8 0 1 2 3 4 5 6 7

6 0 6 12 18 24 30 36 42

Residues 0 6 4 2 0 6 4 2

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Multiplicative Inverse

If p is a prime number, then all the elements of Zp are relatively prime to p Multiplicative inverse (w-1)

For each there exists a z such that w z 1 mod p For each and gcd(w,n)=1, there exists a z such that w

z 1 mod n

Z8 0 1 2 3 4 5 6 7

5 0 5 10 15 20 25 30 35

Residues 0 5 2 7 4 1 6 3

pZw

nZw

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Modulo 8 Example (1)

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Modulo 8 Example (2)

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Properties of Modular Arithmetic for Integer Zn

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Greatest Common Divisor (GCD) A common problem in number theory GCD (a,b) of a and b is the largest number

that divides evenly into both a and b eg GCD(60,24) = 12

Often want no common factors (except 1) and hence numbers are relatively prime eg GCD(8,15) = 1 hence 8 & 15 are relatively prime

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Euclid's GCD Algorithm An efficient way to find the GCD(a,b) uses theorem that:

GCD(a,b) = GCD(b, a mod b) gcd(55,22)=gcd(22,55 mod 22)=gcd(22,11)=11

Euclid's Algorithm to compute GCD(a,b): EUCLID(a,b)1. A a; B b2. If B=0 return A=gcd(a,b)3. R = A mod B4. A B5. B R6. goto 2

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Example GCD(1970,1066)

1970 = 1 x 1066 + 904 gcd(1066, 904)1066 = 1 x 904 + 162 gcd(904, 162)904 = 5 x 162 + 94 gcd(162, 94)162 = 1 x 94 + 68 gcd(94, 68)94 = 1 x 68 + 26 gcd(68, 26)68 = 2 x 26 + 16 gcd(26, 16)26 = 1 x 16 + 10 gcd(16, 10)16 = 1 x 10 + 6 gcd(10, 6)10 = 1 x 6 + 4 gcd(6, 4)6 = 1 x 4 + 2 gcd(4, 2)4 = 2 x 2 + 0 gcd(2, 0)

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Galois Fields Finite fields play a key role in cryptography Can show number of elements in a finite field

must be a power of a prime pn

Known as Galois fields Denoted GF(pn) In particular often use the fields:

GF(p) GF(2n)

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Galois Fields GF(p) GF(p) is the set of integers {0,1, … , p-1}

with arithmetic operations modulo prime p These form a finite field

since have multiplicative inverses Hence arithmetic is “well-behaved” and can

do addition, subtraction, multiplication, and division without leaving the field GF(p)

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Example GF(7) -- (1)

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Example GF(7) -- (2)

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Finding Inverses (1) Can extend Euclid’s algorithm:

EXTENDED EUCLID(m, b)1. (A1, A2, A3)=(1, 0, m); (B1, B2, B3)=(0, 1, b)2. if B3 = 0

return A3 = gcd(m, b); no inverse3. if B3 = 1

return B3 = gcd(m, b); B2 = b–1 mod m4. Q = A3 / B35. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)6. (A1, A2, A3)=(B1, B2, B3)7. (B1, B2, B3)=(T1, T2, T3)8. goto 2

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Finding Inverses (2)

321

321

321

BbBmB

AbAmA

TbTmT

mbB

mBbB

bBmB

BbBmB

mod 1

1

1

2

12

21

321

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Inverse of 550 in GF(1759)

17591650

550545

3

109

5

5

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Polynomial Arithmetic Ordinary polynomial arithmetic

A polynomial with degree n

n

i

ii

nn

nn xaaxaxaxaxf

001

11 ...)(

011110

0

0 1

00

...

where,)()(

)()()(

,)( ,)(

babababac

xcxgxf

xaxbaxgxf

mnxbxgxaxf

kkkkk

mn

i

ii

m

i

n

mi

ii

iii

m

i

ii

n

i

ii

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Polynomial Arithmetic with Coefficients in Zp Polynomial ring Example of GF(2)

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Example of GF(2)

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Irreducible A polynomial f(x) over a field F is called irreducible

if and only if f(x) cannot be expressed as a product of two polynomials.

The polynomial over GF(2) is reducible because

1)( 4 xxf

)1)(1(1 234 xxxxx

13 xx is irreducible

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Finding the GCD EUCLID Algorithm

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Finite Fields of the Form GF(2n) To work with integers that fit exactly into a given

number of bits, with no wasted bit patterns. (for implementation efficiency)

Arithmetic in GF(23) Addition

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Arithmetic in GF(23) Multiplication

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Arithmetic in GF(23) Additive and multiplicative inverses

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Modular Polynomial Arithmetic Consider the set S of all polynomials of degree n-1

or less over the field Zp. Thus, each polynomial has the form

where each ai takes on a value in the set {0,1,…,p-1}. There are a total of pn different polynomials in S.

1

1

)(n

i

ii xaxf

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Arithmetic Operations Arithmetic follows the ordinary rules of polynomial

arithmetic using the basic rules of algebra, with the following refinements.

Arithmetic on the coefficients is performed modulo p. That is, we use the rules of arithmetic for the finite field Zp.

If multiplication results in a polynomial of degree greater than n-1, than the polynomial is reduced modulo some irreducible polynomial m(x) of degree n. That is, we divide by m(x) and keep the remainder. For a polynomial f(x), the remainder is expressed as

r(x)=f(x) mod m(x).

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Example of GF(28) – in AES (1)

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Example of GF(28) – in AES (2)

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Construction of GF(23) Two irreducible

polynomials in GF(23)

1

13

23

xx

xx

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Polynomial Arithmetic Modulo (1)13 xx

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Polynomial Arithmetic Modulo (2)13 xx

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Finding the Multiplicative Inverse

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Implementation Considerations (1) Addition

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Implementation Considerations (2) Multiplication (1)

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Implementation Considerations (3) Multiplication (2)

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Implementation Considerations (4) Multiplication (3)

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AES (Advanced Encryption Standard) Next generation encryption standard of

NIST/FIPS It will replace the use of DES in the following

30 years The sensitive information protected by AES

can not be revealed within 100 years It is selected by the competition from

international selection process

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Calendar of AES

Announcement January 1997 Requirements workshop April 1997 Final requirements September 1997 Pre-submission April 15,1998 Submission June15,1998 AES conference 1-presentation August 20-22,1998 AES conference 2-analysis March 22-23,1999 Selection of 5 finalists April 15,1999 AES conference 3 Beginning of 2000? Final AES selection October 2, 2000

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AES Requirements Block cipher 128-bit block 128/192/256-bit keys It is equal to Triple DES at least on security and is

more efficient Provide descriptions and analysis Provide three implementations in two languages

(reference and optimized in C ， optimized in Java) IF selected, royalty free world wide

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The 15 Submission for AES (1)

Cipher

Submitted

Country

CAST-256 Entrust Canada Crypton Future Korea

Deal Outerbridge Canada DFC ENS-CNRS France E2 NTT Japan

Frog TecApro Costa Rica HPC Schroeppel USA

LOKI97 Brown,Pieprzyk,Seberry Australia

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The 15 Submission for AES (2)

Magenta Deutsche Germany * Mars IBM USA * RC6 RSA USA

* Rijndael Daeman,Rijmen Belgium Safer+ Cylink USA

* Serpent Anderson，Biham ，Kundsen

UK，Israel ，Norway

* Twofish Counterpane USA

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Final AES Selection Rijndael

Block cipher with block size 128 bits Accept 128-, 192-, 256-bit length keys Easy to implement in H/W

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The Implementation of Crypto Algorithms (W32)

http://www.cryptosoft.com Different platforms: win16, win32, linux, OS/2,… Triple DES, Rijndael, Safer+, Blowfish, Cast-128, …

Crypto++: a C++ Class Library of Cryptographic Primitives Version 3.0 1/1/1999 http://www.eskimo.com/~weidai/cryptlib.html

Microsoft CryptoAPI

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More AES Information NIST AES Homepage

http://csrc.nist.gov/encryption/aes/ Rijndael Specification Those who are interested in the

AES specification (i.e., what will be in the standard) should refer to the Draft FIPS for the AES.

Test Values Supporting Documentation Rijndael Developers' Contact Information Rijndael Code: C/C++/Java/Visual Basic

Rijndael Homepage http://www.esat.kuleuven.ac.be/~rijmen/rijndael/

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The AES Cipher AES Parameters

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The AES Cipher - Rijndael Designed by Rijmen-Daemen in Belgium 128/192/256 bit keys, 128 bit data An iterative rather than feistel cipher

treats data in 4 groups of 4 bytes operates an entire block in every round

Designed to be: resistant against known attacks speed and code compactness on many CPUs design simplicity

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Rijndael Processes data as 4 groups of 4 bytes (state) Steps

byte substitution (uses an S-box to perform a byte-by-byte substitution of the block)

shift rows (a simple permutation) mix columns (substitution uses arithmetic over GF(28)) add round key (a simple bitwise XOR of the current block

with a portion of the expanded key) All operations can be combined into XOR and table

lookups - hence very fast & efficient

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Rijndael

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AES Data Structure

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Byte Substitution A simple substitution of each byte Uses one table of 16x16 bytes containing a

permutation of all 256 8-bit values Each byte of state is replaced by byte in row (left 4-

bits) & column (right 4-bits) eg. byte {95} is replaced by row 9 col 5 byte which is the value {2A}

S-box is constructed using a defined transformation of the values in GF(28)

Designed to be resistant to all known attacks

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Example of the SubBytes

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S-box of AES (1)

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S-box of AES (2)

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AES Byte-Level Operations (1)

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AES Byte-Level Operations (2)

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Construction of the S-box (1)

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Construction of the S-box (2)

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Construction of the S-box (3)

{95}-1 in GF(28) = {8A} = {10001010}

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Construction of the S-box (4) Inverse substitute byte transformation

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Construction of the S-box (5)

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Shift Rows (1) A circular byte shift in each each

1st row is unchanged 2nd row does 1 byte circular shift to left 3rd row does 2 byte circular shift to left 4th row does 3 byte circular shift to left

Decrypt does shifts to right Since state is processed by columns, this step

permutes bytes between the columns

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Shift Rows (2)

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Mix Columns (1) Each column is processed separately Each byte is replaced by a value dependent on

all 4 bytes in the column Effectively a matrix multiplication in GF(28)

using prime polynomial m(x) =x8+x4+x3+x+1

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Mix Columns (2)

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Example of the MixColumns (1)

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Example of the MixColumns (2)

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Inverse MixColumns (1)

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Inverse MixColumns (2)

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Add Round Key XOR state with 128-bits of the round key Again processed by column (though

effectively a series of byte operations) Inverse for decryption is identical since XOR

is own inverse, just with correct round key Designed to be as simple as possible

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AES Round

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Example of Add Round Key

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AES Key Expansion Takes 128-bit (16-byte) key and expands into

array of 44/52/60 32-bit words Start by copying key into first 4 words Then loop creating words that depend on

values in previous & 4 places back in 3 of 4 cases just XOR these together every 4th has S-box + rotate + XOR constant of

previous before XOR together Designed to resist known attacks

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Algorithm (1)

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Algorithm (2)

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Example of AES Key Expansion

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AES Decryption (1) AES decryption is not identical to encryption

since steps done in reverse But can define an equivalent inverse cipher

with steps as for encryption but using inverses of each step with a different key schedule

Works since result is unchanged when swap byte substitution & shift rows swap mix columns & add (tweaked) round key

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AES Decryption (2) Equivalent Inverse

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Implementation Aspects Can efficiently implement on 8-bit CPU

byte substitution works on bytes using a table of 256 entries

shift rows is simple byte shifting add round key works on byte XORs mix columns requires matrix multiply in GF(28)

which works on byte values, can be simplified to use a table lookup

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Implementation Aspects Can efficiently implement on 32-bit CPU

redefine steps to use 32-bit words can precompute 4 tables of 256-words then each column in each round can be computed

using 4 table lookups + 4 XORs at a cost of 4*(1024 bytes) to store tables

Designers believe this very efficient implementation was a key factor in its selection as the AES cipher