Cryptography

Electronic Codebook Book (ECB)
• message is broken into independent
blocks which are encrypted
• each block is a value which is substituted,
like a codebook, hence name
• each block is encoded independently of
the other blocks
Ci = DESK1 (Pi)
• uses: secure transmission of single values

Electronic Codebook Book (ECB)

Advantages and Limitations of ECB
• repetitions in message may show in
ciphertext
– if aligned with message block
– particularly with data such graphics
– or with messages that change very little,
which become a code-book analysis problem
• weakness due to encrypted message
blocks being independent
• main use is sending a few blocks of data

Cipher Block Chaining (CBC)
• message is broken into blocks
• but these are linked together in the
encryption operation
• each previous cipher blocks is chained
with current plaintext block, hence name
• use Initial Vector (IV) to start process
Ci = DESK1(Pi XOR Ci-1)
C-1 = IV
• uses: bulk data encryption, authentication

Advantages and Limitations of CBC
• each ciphertext block depends on all message blocks
• thus a change in the message affects all ciphertext
blocks after the change as well as the original block
• need Initial Value (IV) known to sender & receiver
– however if IV is sent in the clear, an attacker can change bits of
the first block, and change IV to compensate
– hence either IV must be a fixed value (as in EFTPOS) or it must
be sent encrypted in ECB mode before rest of message
• at end of message, handle possible last short block
– by padding either with known non-data value (eg nulls)
– or pad last block with count of pad size
• eg. [ b1 b2 b3 0 0 0 0 5] <- 3 data bytes, then 5 bytes pad+count

Cipher FeedBack (CFB)
• message is treated as a stream of bits
• added to the output of the block cipher
• result is feed back for next stage (hence name)
• standard allows any number of bit (1,8 or 64 or
whatever) to be feed back
– denoted CFB-1, CFB-8, CFB-64 etc
• is most efficient to use all 64 bits (CFB-64)
Ci = Pi XOR DESK1(Ci-1)
C-1 = IV
• uses: stream data encryption, authentication

Advantages and Limitations of CFB
• appropriate when data arrives in bits/bytes
• most common stream mode
• limitation is need to stall while do block
encryption after every n-bits
• note that the block cipher is used in
encryption mode at both ends
• errors propogate for several blocks after
the error

Output FeedBack (OFB)
• message is treated as a stream of bits
• output of cipher is added to message
• output is then feed back (hence name)
• feedback is independent of message
• can be computed in advance
Ci = Pi XOR Oi
Oi = DESK1(Oi-1)
O-1 = IV
• uses: stream encryption over noisy channels

Advantages and Limitations of OFB
• used when error feedback a problem or where need to
encryptions before message is available
• superficially similar to CFB
• but feedback is from the output of cipher and is
independent of message
• a variation of a Vernam cipher
– hence must never reuse the same sequence (key+IV)
• sender and receiver must remain in sync, and some
recovery method is needed to ensure this occurs
• originally specified with m-bit feedback in the standards
• subsequent research has shown that only OFB-64
should ever be used

Counter (CTR)
• a “new” mode, though proposed early on
• similar to OFB but encrypts counter value
rather than any feedback value
• must have a different key & counter value
for every plaintext block (never reused)
Ci = Pi XOR Oi
Oi = DESK1(i)
• uses: high-speed network encryptions

Advantages and Limitations of CTR
• efficiency
– can do parallel encryptions
– in advance of need
– good for bursty high speed links
• random access to encrypted data blocks
• provable security (good as other modes)
• but must ensure never reuse key/counter
values, otherwise could break (cf OFB)

Summary
• block cipher design principles
• DES
• Differential & Linear Cryptanalysis
• Modes of Operation
– ECB, CBC, CFB, OFB, CTR

Finite Fields
• Important in cryptography
– AES, Elliptic Curve, IDEA, Public Key

• Groups, rings, fields from abstract algebra

Group
• a set of elements or “numbers”
• with some operation whose result is also
in the set (closure)
• obeys:
– associative law: (a.b).c = a.(b.c)
– has identity e: e.a = a.e = a
– has inverses a-1: a.a-1 = e
• if commutative a.b = b.a
– then forms an abelian group

Cyclic Group
• define exponentiation as repeated
application of operator
– example: a-3 = a.a.a
• and let identity be: e=a0
• 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
• a is said to be a generator of the group

Ring
• a set of “numbers” with two operations (addition
and multiplication) which are:
• an abelian group with addition operation
• multiplication:
– has closure
– is associative
– distributive over addition: a(b+c) = ab + ac
• if multiplication operation is commutative, it
forms a commutative ring
• if multiplication operation has inverses and no
zero divisors, it forms an integral domain

Field
• a set of numbers with two operations:
– abelian group for addition
– abelian group for multiplication (ignoring 0)
– ring

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 same remainder
– eg. 100 = 34 mod 11
• b is called the residue of a mod n
– since with integers can always write: a = qn + b
• usually have 0 <= b <= n-1
-12 mod 7 ≡ -5 mod 7 ≡ 2 mod 7 ≡ 9 mod 7

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
...

Divisors
• say a non-zero number b divides a if for
some m have a=mb (a,b,m all integers)
• that is b divides into a with no remainder
• denote this b|a
• and say that b is a divisor of a
• eg. all of 1,2,3,4,6,8,12,24 divide 24

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

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

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

Euclid's GCD Algorithm
• an efficient way to find the GCD(a,b)
• uses theorem that:
– GCD(a,b) = GCD(b, a mod b)
• Euclid's Algorithm to compute GCD(a,b):
– A=a, B=b
– while B>0
• R = A mod B
• A = B, B = R
– return A

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)

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)

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)

Finding Inverses
• 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 inverse
3. if B3 = 1
return B3 = gcd(m, b); B2 = b–1 mod m
4. Q = A3 div B3
5. (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

Polynomial Arithmetic
• can compute using polynomials

• several alternatives available
– ordinary polynomial arithmetic
– poly arithmetic with coords mod p
– poly arithmetic with coords mod p and
polynomials mod M(x)

Ordinary Polynomial Arithmetic
• add or subtract corresponding coefficients
• multiply all terms by each other
• eg
– let f(x) = x3 + x2 + 2 and g(x) = x2 – x + 1
f(x) + g(x) = x3 + 2x2 – x + 3
f(x) – g(x) = x3 + x + 1
f(x) x g(x) = x5 + 3x2 – 2x + 2

Polynomial Arithmetic with Modulo
Coefficients
• when computing value of each coefficient
do calculation modulo some value
• could be modulo any prime
• but we are most interested in mod 2
– ie all coefficients are 0 or 1
– eg. let f(x) = x3 + x2 and g(x) = x2 + x + 1
f(x) + g(x) = x3 + x + 1
f(x) x g(x) = x5 + x2

Modular Polynomial Arithmetic
• can write any polynomial in the form:
– f(x) = q(x) g(x) + r(x)
– can interpret r(x) as being a remainder
– r(x) = f(x) mod g(x)
• if have no remainder say g(x) divides f(x)
• if g(x) has no divisors other than itself & 1
say it is irreducible (or prime) polynomial
• arithmetic modulo an irreducible
polynomial forms a field

Polynomial GCD
• can find greatest common divisor for polys
– c(x) = GCD(a(x), b(x)) if c(x) is the poly of greatest
degree which divides both a(x), b(x)
– can adapt Euclid’s Algorithm to find it:
– EUCLID[a(x), b(x)]
1. A(x) = a(x); B(x) = b(x)
2. 2. if B(x) = 0 return A(x) = gcd[a(x), b(x)]
3. R(x) = A(x) mod B(x)
4. A(x) ¨ B(x)
5. B(x) ¨ R(x)
6. goto 2

Modular Polynomial Arithmetic
• can compute in field GF(2n)
– polynomials with coefficients modulo 2
– whose degree is less than n
– hence must reduce modulo an irreducible poly
of degree n (for multiplication only)
• form a finite field
• can always find an inverse
– can extend Euclid’s Inverse algorithm to find

Computational Considerations
• since coefficients are 0 or 1, can represent
any such polynomial as a bit string
• addition becomes XOR of these bit strings
• multiplication is shift & XOR
– cf long-hand multiplication
• modulo reduction done by repeatedly
substituting highest power with remainder
of irreducible poly (also shift & XOR)

Cryptography

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Cryptography