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Cryptography

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Barry Coggins

on 31 January 2017

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Transcript of Cryptography

Cryptography &
Math
Hiding In Plain Sight
Barry Coggins
sluf@sluf.com
Certified Information System Security Professional
Mississippi School for Mathematics and Science
Jsccsccsrrs Cuiddb tdw Jgxihjgxsuc gkq Cushkuh
Mississippi School for Mathematics and Science
01001101011010010111001101110011011010010111001101110011011010010111000001110000011010010010000001010011011000110110100001101111011011110110110000100000011001100110111101110010001000000100110101100001011101000110100001100101011011010110000101110100011010010110001101110011001000000110000101101110011001000010000001010011011000110110100101100101011011100110001101100101
secret
011100110110010101100011011100100110010101110100
XOR
00111110000011000001000000000001000011000000011100000000000011000001001100000010000011000101010000100000000001100000101100011101000010100001100001010011000000110000110000000000010001010011100100010010000100010000101100010111000010000001010100000111000011000000000000000001010001010001010100011101000000010100001100100001000001100001110100010110000010110000000000010111
Symmetric Asymmetric
Lock & Unlock
Lock
Unlock
Mathematically Related
Pick 2 Prime Numbers:

Example:
p = 11
q = 17
Multiply them together:

n = p * q
n = 11 * 17
n = 187
Compute Euler's totient function of n:

φ(n) = (p - 1) * (q - 1)
φ(187) = (11 - 1) * (17 - 1)
φ(187) = (10) * (16)
φ(187) = 160
Pick a random number e such that
1 < e < φ(n)
1 < e < φ(187)
1 < e < 160
*and*
e is a coprime with φ(n) meaning no common factors
e = 27
Compute d, the modular multiplicative inverse of e (mod φ(n))
e^-1 = d (mod φ(n))
27^-1 = d (mod φ(187))
27^-1 = d (mod 160)
27 * d = 1 (mod 160)
d = 83
(source: http://planetcalc.com/3311/)
public key = (n = 187, e = 27)
private key = (n = 187, d = 83)

Secret Message ("M" = 77)
c(m) = m^e mod n
c(77) = 77^27 mod 187
c(77) = 66

Decrypt Message (received 66)
m(c) = c^d mod n
m(66) = 66^83 mod 187
m(66) = 77
For further reading:
Diffe-Hellman Key Exchange
Elliptic Curve Crypto
RSA Algorithm
Public Key Infrastructure
Very Large Prime Numbers
>,0!,' ,3",T &+=*8S#, E921+7(5',

F8304004301C00313020C5420060B13428614C0C3001143912110B17081507300005145474050321061D160B0017
Galois Counter Mode (GCM)
Questions?
"That Depends"

See these slides at:
https://go.sluf.com/MSMS.20170131
Effective Encryption
Share my video with Alice
Symmetric Key
Encrypt Symmetric Key
with Alice's Public Key
Decrypt with the Private Key to reveal the Symmetric Key
Randomness
Entropy
Random data to "mix" your plaintext message
If your password is "monkey", you're wrong
Lavarand
https://en.wikipedia.org/wiki/Lavarand
n = 11 * 17
e = random value *
d = multiplicative inv
Difficult to factor large prime numbers
Tease the Math (ElGamal)
Bob generates P, G, x
Y = G^x mod P
- Y, G, P are the public key

Alice generates k and message M
- a = G^k mod P
- b = Y^k * M mod P

Bob decrypts M
- M(plain) = b / (a^x) mod P
Difficult to solve the discrete logarithm
Tease the Math (ECC)
Difficult to solve the elliptic curve
y^2 = x^3 + ax + b
Sluf Consulting
Security & Compliance
Use REALLY BIG numbers
10^8 atoms in the known universe
2^2024 bits of encryption strength
2^270 is big
2^271 is doubled
2^272 is doubled again
Full transcript