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# Cryptography

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by

Tweet## Barry Coggins

on 31 January 2017#### 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 transcriptMath

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