By Boris Ryabko

ISBN-10: 9812564055

ISBN-13: 9789812564054

ISBN-10: 9812703306

ISBN-13: 9789812703309

The purpose of this booklet is to supply a finished advent to cryptography with no utilizing advanced mathematical buildings. the subjects are conveyed in a sort that in simple terms calls for a simple wisdom of arithmetic, however the tools are defined in enough aspect to permit their machine implementation.

The publication describes the most thoughts and amenities of latest cryptography, proving key effects alongside the way in which. The contents of the 1st 5 chapters can be utilized for one-semester path.

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**Additional info for Basics of Contemporary Cryptography for IT Practitioners**

**Sample text**

In this case encryption is maximally fast (it requires only 2 modular multiplications). 17 Suppose Alice wants to send Bob the message m = 15. Let Bob’s parameters be PB = 3, QB = 11, N B = 33, d B = 3 ) 20). Find cg using the extended Euclidean algo(3 is coprime to ( ~ ( 3 3= rithm: C B = 7 (check it: 3 . 7 mod 20 = 1). Encrypt m using Eq. 15 mod 33 = 9 . Alice sends the number 9 to Bob over an open channel. 9’ . 15 ’ 9 mod 33 = 15. We can see that Bob has deciphered the message. 30 Basics of Contemporary Cryptography for IT Practitioners The considered system is unbreakable if P and Q are large but has the following imperfection: A sends a message to B by utilising B’s public information (the numbers NB and d B ) .

17) Substituting the right of Eq. 17) for s-l in Eq. 16) we obtain = ( a l r b ( h + z r ) - ' a 2 r k ( ~ + z r ) - ' modp modq ) = (a(h+r"-l(h+Zr)k mod p ) mod q = (ak mod p ) mod q . Taking into account the definition of r (Step 3) we conclude that which completes the proof. 13) the following method is recommended. Select a random integer g > 1 and compute a = g ( p - ' ) / q mod p . If a > 1 then it is what we need. e. 13) holds. 18) we obtain a = 1 (extremely improbable case) then we should try another value of g.

Lo4 mod 47 = 3 7 . 3 6 mod 47 = 16 = 2 . 2 . 2 . 2 . The number 16 is 5-smooth and the fourth step is completed. Turn to logarithms in the last equality (this is the fifth step) and obtain the final result: 10g1037 = 410g102 - 4 = ( 4 . 3 0 - 4) mod 46 = 2 4 . We have found the solution of Eq. 14) x = 24. We may check it: 47 = 37. mod The fastest to the date is the variant of the described index calculus algorithm called Number Field Sieve, see [Lenstra and Lenstra (1993)l. This method is based upon subtle algebraic constructions so we do not describe it in this book.

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