Information Security Seminar #2 for 4th course of radiotechics faculty of MIPT

1. Seminar #2
Infosecurity
Number theory. Group, ring and
field theory
Kolybelnikov Alexander
kisttan@gmail.com

2. Agenda
•Number theory
–Modular arithmetic
–GCD (greatest common divisor)
• Fermat theorem, Euler function, modulo
inverse value
• Group, ring, field theory

3. Number theory
Modular arithmetic

4. Modular arithmetic. Definitions
•Two integer numbers a and b are comparable
in absolute value n (modulo n parity) if a and b
give the same remainder of division by n.
•Example: 32 and −10 are comparable in
absolute value 7 because 32 = 7∙4 + 4, −10 =
7∙(-2) + 4.
•Statement «a and b are comparable in absolute
value n» is commonly expressed as:
a ≡b (mod n)

5. Modular arithmetic.
Characteristics
•Comparison relation means a relation of
equivalence and is characterized by many
properties of common equations.
•For instance, addition and multiplication are
allowed in following cases:
a2≡b2(mod n)
•a1≡b1(mod n)
•a1 a2 ≡b1b2 (mod n) a1 +a2 ≡b1+b2 (mod n)
•
•
•

6. Prime numbers and methods of their
detection

7. Prime numbers
•Prime number is a natural number that has
exactly two different natural divisors: unit and
itself. All other numbers except unit element are
called composite numbers. So all natural
numbers greater than unit are divided into prime
and composite. Number theory studies properties
of prime numbers. In ring theory irreducible
elements correspond to prime numbers.

8. Tests detecting prime numbers
•Sieve of Eratosthenes
•Fermat test
•For the next seminars:
–Miller Rabin test
–AKS test

9. Sieve of Eratosthenes
•Choosing of a random number N;
•Dividing of N by all numbers from 1 to N/2.

10. Fermat test
•If p is a prime number then
• a p − 1 ≡ 1 (mod p)
•The probability that p is a prime number
increases with large amount of testing a and
equals to 1/2n , n means amount of a.
•
•

11. Prime numbers properties
•If p is a prime number and p is divisible by ab then p is divisible bya and by b.
(Euclidean lemma, this is used in the proof of fundamental theorem of
arithmetic.
•Residue ring is considered to be a field if and only if n is a prime number.
•Field characteristic is zero or a prime number.
•If p is a prime number and a is a natural number then (ap − a) is divisible by p
(Fermat little theorem).
•If G is a finite group of pn elements then G contains an element of p order.
•If G is a finite group and pn is the maximum power of p that divides | G | then
G has a subgroup of pn order (Sylov subgroup) and the number of Sylov
subgroups is equal to pk + 1 for an integer k (Sylov's theorems).
•Natural number p > 1 is a prime number if and only if ((p − 1)! + 1) is
divisible by p (Vilson theorem).
•If n > 1 is a natural number then a prime number p is defined as follows:
•n < p < 2n (Bertrand postulate).

12. Greatest common divisor (GCD)

13. GCD. Definition
Greatest common divisor (GCD) for two
integer numbers m and n is the greatest from
their common divisors.
Example: for numbers 70 and 105 GCD = 35

14. GCD calculation. Euclidean
algorithm
•Provides a and b are integer numbers that are never both zero together
and number sequence
•
is defined in the way that each rk is a remainder of division of previous
but one by previous number
and the last but one number is divisible by the last number (without a
remainder):
a = bq0 + r1 b = r1q1 + r2 r1 = r2q2 + r3 rk − 2 = rk − 1qk − 1 + rk rn − 1 = rnqn Then
GCD (a,b) = rn ; rn is the last non-zero member of the sequence.
•The presence of r1,r2,..(that means possibility of division of m by n with
a remainder for any integer m and n) is proved by induction on m.
•The following statement: if a = bq + r then GCD(a,b) = GCD(b,r) leads
to the correctness of this algorithm.

15. Inverse element calculation
•Input: integer numbers а, b: 0 < b ≤ а.
•Output: d = GCD(а, b); integer numbers х, у: ах+ by = d.
•1. Initiate r0 ← а, r1 ← b, х0 ← 1, x1 ← 0, у0 ← 0, y1 ←1, i ← 1
•2. Divide ri–1 by ri with a remainder: ri–1 = qiri +ri+1
•3. If ri+1 = 0 then initiate d ← ri , х ← xi у ← yi. Else initiate xi+1 ← xi–1 –
qixi, yi+1 ← yi–1 – qiyi, i← i + 1 and go to step 2.
•4. Result: d, х, у.

16. Extended Euclidean algorithm
•Is used for:
–GCD calculation;
–Modulo inverse element calculation.
•Example:
–calculate GCD (462, 1071);
–Calculate modulo inverse element for 109 in Galois
field GF(28).

17. Example# 1 – GCD calculation
•Calculate GCD (462, 1071)
•1. Subtract a multiple value of 462 from
1071 before a remainder is less than 462.
q0 = 2, remainder = 147
1071 = 2 × 462 + 147
•2. Subtract a multiple value of 147 from 462
before a remainder is less than 147. q1 = 3,
remainder = 21.
462 = 3 × 147 + 21

18. Example# 1 – GCD calculation
•3. Subtract a multiple value of 21 from 147
before a remainder is less than 21.
•q2 = 7, remainder = 0.
147 = 7 × 21 + 0.
The last remainder is equal to 0,
subsequently, the algorithm output is 21;
GCD(1071, 462)=21.

19. Example#2 — Modulo inverse
element calculation
•Given: a = 1769, b = 551.
•Using extended Euclidean algorithm find
integer numbers x и y: d = ax + by,
•d = GCD(a, b).
•

20. Example#2 — Modulo inverse
element calculation
1. Initiate a0 = 1, a1 = 0, b0 = 0, b1 = 1, a= 1769, b =
551.
2. Quotient q = a/b = 1769/551 = 3, remainder r =
116.
3. If r = 0 then go to the last step (6).
4. Initiate: a = 551; b = 116; t = a0 = 1; a0 = a1 =0;
a1 = t – a1q = 1 – 0 · = 1 t = b0 = 0; b0 = b1 = 1; b1
= t – b1q = – 3;
Result of the step : a= 551, b = 116, a0 = 0, a1 = 1,

21. Example#2 — Modulo inverse
element calculation
1. Parameters values: a = 551, b = 116, a0 = 0, a1 =
1, b0 = 1, b1 = – 3.
2. Quotient q = a/b = 551/116 = 4, remainder r =
87.
3. If r = 0 then go to the last step (6).
4. Initiate: a = 116; b = 87; t = a0 = 0; a0 = a1 =1; a1
= t – a1q = 0 – 1 · 4 = – 4t = b0 = 1;b0 = b1 = – 3;
b1 = t – b1q = 1 – ( – 3) 4 = 13;
Result of the step : a= 116, b = 87, a0 = 1, a1 = – 4,
b = – 3, b = 13.

22. Example#2 — Modulo inverse
element calculation
1. Parameters values: a=116, b =87, a0 =1, a1 = –4, b0
= –3, b1=13
2. Quotient q = a/b = 116/87 = 1, remainder r = 29.
3. If r = 0 then go th the last step (6).
4. Initiate: a = 87; b = 29; t = a0 = 1; a0 = a1 = – 4: a1 =
t – a1q = 1 – ( – 4) · 1 = 5; t = b0 = – 3; b0 = b1 = 13; b1
= t – b1q = – 3 – (13) · 1 = – 16;
Result of the step: a= 87, b = 29, a0 = – 4, a1 = 5, b0 =
13, b1 = – 16.

23. Example#2 — Modulo inverse
element calculation
1. Parameters values: a= 87, b = 29, a0 = –4, a1 =5, b0
=13, b1 = –16.
2. Quotient q = a/b = 87/29 = 3, remainder r = 0.
3. If r = 0 then go to the last step (6).
4. Initiate: a = 87; b = 29; t = a0 = – 4; a0 = a1 = 5; – 4
– 5·3 = – 19; t = b0 = 13;
b0 = b1 = – 16; b1 = t – b1q = 13 – ( – 16)·3 = 61;
Result of the step: a= 87, b = 29, a0 = 5, a1 = – 19,
b0 = – 16, b1 =6

24. Result
Calculate GCD according to formula d = ax + by,
x = x0 = 5, y = y0 = – 16, a= 1769, b = 551.
According to parameters values:
d = ax + by = 1769 · 5 – 551 · 16 = 8845 – 8816 =29.

25. Fermat little theorem
•If p is a prime number and integer number a is
not divisible by p then
•a p − 1 ≡ 1 (mod p) (a p − 1 − 1 is divisible by p).

26. Euler function
•Euler function
(n is a natural number)
is equal to a number of natural numbers that
are non-greater than n and are coprime with n.
It is named after Euler who was the first used
it in his works on number theory.

27. Group, ring and field theory.
Brief information

28. Group. Definition
•Group is a nonempty multitude with
defined binary operation.
•Axioms for a group:
•Associativity:(a*b)*c=a*(b*c)
•The presence of neutral element:
e*a=a*e=a
•The presence of inverse element:
•a*a-1=a-1*a=e

29. Ring
•Ring is a nonempty multitude with two binary
operations defined on it.
•Axioms for ring:
•Addition commutativity: a+b=b+a
•Addition associativity: (a+b)+c=a+(b+c)
•Addition neutral element: a+0=0+a=a
•Addition inverse element: a+b=b+a=0
•Multiplication associativity a*(b*c)=(a*b)*c
•Multiplication commutativity: a*b=b*a
•Distributivity: a*(b+c)=a*b+a*c
•

30. Field of numbers
Field is a commutative, associative ring with
unit element and reversible non-zero
elements.

31. Galois fields
•Finite field (Galois field) is a field that contains
finite number of elements.
•Finite field is commonly expressed as GF(q), q is a
number of the field elements.
•Simple example of finite field is residue ring
modulo a prime number p.

32. Galois fields
•Field GF(pn) defining (p is a prime number, n is a natural number) starts with defining of its
prime subfield GF(p) (that is equal to the whole field for n=1).
•Prime field GF(p) is defined as a residue ring modulo p, that doesn't have zero divisors and
is a field (because p is a prime number).
•Elements are numbers. Operate with elements like integer numbers, reducting a result on p
modulo.
•Field GF(pn) for n>1 is defined as a factor ring, f(x) is a prime polynomial of degree n on
the field. So to define a field from pn elements it is enough to find a polynomial of degree n
that is a prime for the field.
•Field elements are polynomials of degree less than n. Arithmetic operations (addition and
multipliciation) are on modulo of polynomial f(x), so the result corresponding to an
operation is a remainder from division by f(x) with reduction of coefficients on modulo p.

33. Literature
•Number theory. Vinogradov I. M.
•Groups, rings, fields. Zelvensky I.G.
SPbGETU