CRYPTOGRAPHY
Presented by:
Noushin Ranjkesh
Olinka Bedroya
Sharif University of Technology-1391
Department of Physics, Tehran, Iran
CHAPTER 4
THE LANGUAGE BARRIER
The impenetrability of
unknown languages, the Navajo
code talkers of World War II
and the decipherment of
Egyptian hieroglyphs
PURPLE CODE ,
BATTLE OF MIDWAY
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‫بیهودگی بنیادی ماشین های رمزنگاری‪ ،‬سرعت پایین‬
‫انتقال اطالعات در نبردهای مناطق محدود‪ ،‬به ویژه‬
‫جنگ های اقیانوس آرام‪ ،‬در نهایت موجب استفاده از‬
‫کد گوهای ناواجو گردید‪.‬‬
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‫کاستی های استفاده از زبان ناواجو برای‬
‫رمزگذاری اطالعات در جنگ جهانی دوم‬
‫‪ ‬نبودن واژه های معادل در زبان ناواجو‪،‬برای واژه های نظامی‬
‫معادل سازی واژه های کاربردی در جنگ با نام حیوانات و واژهای‬
‫اصیل در زبان ناواجو‬
‫‪ ‬نام مکان ها و افراد که در هر زبانی‪ ،‬به صورت مشابه ادا می شود‬
‫تهیه ی واژه نامه ای از واژه های دارای معادل‪،‬در زبان ناواجو برای‬
‫هر حرف الفبا‬
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The military terms
fighter plane
 amphibious vehicle
 Submarine

The Navajo terms
owl (Da-he-tih-hi)
 frog (Chal)
 iron fish (Besh-lo)

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PACIFIC
Pig
 Ant
 Cat
 Ice
 Fox
 Ice
 Cat

Bi-sodih
 Wol-la-chee
 Moasi
 Tkin
 Ma-e
 Tkin
 Moasi

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DECIPHERING LOST LANGUAGE
AND ANCIENT SCRIPTS
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HIEROGLYPH
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‫‪HIEROGLYPH‬‬
‫‪ ‬زبان مصریان باستا ن از حدود سه هزار سال پیش از میالد‬
‫‪ ‬یک زبان بسیار زینتی و کاربردی برای معابد‬
‫به دلیل سختی نوشتار این زبان کم کم به زبان هیراتیک و سپس دموتیک‬
‫که برای استفاده در روزمره مناسب تر است‪ ،‬تبدیل شده اند‪.‬‬
‫در حدود چهار قرن پس از میالد مسیح‪،‬با گسترش مسیحیت و افزایش قدرت‬
‫نفوذ کلیسا‪،‬زبان یونانی غالب شد و الفبای جدیدی از ترکیب ‪ 24‬حرف از‬
‫زبان یونانی و ‪ 6‬نشانه از زبان دموتیک ساخته شد و زبان قبطی شکل‬
‫گرفت‪.‬‬
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HIERATIC
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DEMOTIC
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COPTIC
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THE ROSETTA STONE
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‫سنگ روزتا‬
‫‪ ‬توسط گروه باستان شناس اعزامی اسکندر در شهر روزتا کشف شد‪.‬‬
‫‪ ‬در موزه ی بریتانیا نگه داری می شود‪.‬‬
‫‪ ‬شامل یک متن ثابت به سه زبان یونانی‪ ،‬دموتیک و هیروگلیف است‪.‬‬
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THOMAS YOUNG
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KARNAK TEMPLE
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Jean-François Champollion
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Berenika
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Ptolemaios
Cleopatra
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‫‪RA‬‬
‫(درقبطی به خورشید گفته می شود)‬
‫)رامسس (‪RAMSS‬‬
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CHAPTER 5
ALICE AND BOB GO PUBLIC
Modern cryptography,
the solution to the so-called
key-distribution problem
and the secret history of
nonsecret encryption
ENTERING THE COMPUTER AGE
Break of lorenz cypher: sending a same 4000 characters
message twice (slightly different)
Bill Tutte
A Lorenz cypher machine
John Tiltman
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ENTERING THE COMPUTER AGE
Max Newman
Colossus,delivered 1943
ENIAC ,1945
Thomas H. Flowers
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ENTERING THE COMPUTER AGE
Differences between computer and mechanical encryption:



complexity
speed
Computers deal with binary numbers
e.g. computer version of a substitution cipher:
Message
Message in ASCII
Key(DAVID)
Ciphertext
HELLO
10010001000101100110010011001001111
10001001000001101011010010011000100
00011000000100001101000001010001011
ASCII table
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ENTERING THE COMPUTER AGE





1947, invention of transistor
1951, Ferranti began to make computers to order.
1953, IBM launched its first computer
1957, introduction of Fortran
1959, invention of the integrated circuit
First transistor
Ferranti’s computer
IBMs first computer
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KEY DISTRIBUTION

Vigenère key

Delivering the Enigma monthly code book

1970s, Banks needed to deliver keys to customers
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KEY DISTRIBUTION
Merkle puzzles
Quadratic gap is best possible if we treat cipher as a black box
oracle
[B. Barak and M. Mahmoody-Ghidary. Merkle Puzzles are Optimal]
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KEY DISTRIBUTION

Whitfield Diffie


connections of the world wide web.
Colors represent different domains.
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KEY DISTRIBUTION

Exchanging keys in person
Martin Hellman

No key sharing - double locked

Asymmetric key…
Ralph Merkle
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ASYMMETRIC KEY
What symmetric and asymmetric indicate
To build an asymmetric cipher:
 Alice publishes a public key
 People lookup for Alice’s Public key
 They use the public key and encryption method to send Alice
messages
 Alice uses her private key to decrypt
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ONE WAY FUNCTIONS
Mixing colors
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ONE WAY FUNCTIONS
Modular Exponentiation
Diffie-Hellman

Agree on a public modulus N and a base g

Alice chooses a private key x between 1 and N -1

She constructs a public key by computing X=g x mod N

Bob chooses a random y and calculate K= X y.

Bob sends Alice the enciphered text and Y=g y

Alice calculates the K= Y x and deciphers the text
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ONE WAY FUNCTIONS
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RSA CRYPTOSYSTEM
Ronald Rivest, Adi Shamir and Leonard Adleman.
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RSA CRYPTOSYSTEM
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RSA CRYPTOSYSTEM







N = 114,381,625,757,888,867,669,235,779,976,146,612,010,
218,296,721,242,362,562,561,842,935,706,935,245,733,897,
830,597,123,563,958,705,058,989,075,147,599,290,026,879,
543,541
q = 3,490,529,510,847,650,949,147,849,619,903,898,133,417,
764,638,493,387,843,990,820,577
p = 32,769,132,993,266,709,549,961,988,190,834,461,413,177,
642,967,992,942,539,798,288,533
Thus far, the best way known to invert RSA is to factor N.
The best running time for a fully proved algorithm is Dixon’s
Random squares which runs in time ~𝑂(exp( log 𝑁 log(log 𝑁 )))
It took 2 years to factor a 232 digit number, using hundreds of
machines
P should have 1024 bits
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THE SECRET HISTORY OF PUBLIC KEY
CRYPTOGRAPHY
James Ellis, joined GCHQ in 1965
Both new and old GCHQbuildings
Malcolm Williamson , joined GCHQ in 1974
Clifford Cocks , joined GCHQ in 1973
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Brief review of chapter 5
•
Birth of computer encryption
•
Key distribution problem
•
Asymmetric encryption
•
Diffie-Hellman cryptosystem
•
RSA crypto system
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APPENDIX
NUMBER THEORY BASIS FOR
RSA CRYPTOSYSTEM
Number Theory Background

𝑎𝑥 ≡ 1 𝑚𝑜𝑑𝑛
gcd 𝑎, 𝑛 = 1

𝜙𝑛 is the number of elements in ℤ𝑛 relatively prime to 𝑛

ℤ𝑛∗ = 𝑎 ∈ ℤ𝑛 | gcd 𝑎, 𝑛 = 1 forms an Abeliangroup
𝑎1 . 𝑎2 . 𝑎3 = 𝑎1 . 𝑎2 . 𝑎3
 𝑎1 . 𝑎2 = 𝑎2 . 𝑎1
 𝑎. 1 = 𝑎
 𝑎. 𝑎 −1 = 1


Theorem: If 𝑏 ∈ ℤ𝑛∗ then 𝑏 𝜙𝑛 ≡ 1 𝑚𝑜𝑑𝑛
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HOW TO FIND PRIME NUMBERS
Deterministic

AKS primality test

Fermat primality test

Miller–Rabin primality test

Solovay-strassen primality test

take a preselected random number of the desired length

apply a Fermat primality test

Probabilistic
apply a certain number of Miller–Rabin tests (depending on the
length and the allowed error rate) to get a number which is very
probably a prime number.
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HOW TO FIND PRIME NUMBERS
AKS primality test

Input: integer n > 1.If n = ab for integers a > 0 and b > 1,
output composite.

Find the smallest r such that or(n) > log2(n).

If 1 < gcd(a,n) < n for some a ≤ r, output composite.

If n ≤ r, output prime.

For a = 1 to do if (X+a)n≠ Xn+a (mod Xr − 1,n), output composite;

Output prime.
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HOW TO FIND PRIME NUMBERS
Fermat primality test

Choose random 𝑝 ∈ 21024

2𝑝−1 ≡ 1

𝑝
If so output p as prime; else go back to first step and choose another
random number
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑎𝑡 𝑝 𝑖𝑠 𝑛𝑜𝑡 𝑝𝑟𝑖𝑚𝑒 < 2−60
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