Algebra for Applications: Cryptography, Secret Sharing, by Arkadii Slinko

By Arkadii Slinko

This e-book examines the connection among arithmetic and information within the glossy global. certainly, smooth societies are awash with information which has to be manipulated in lots of alternative ways: encrypted, compressed, shared among clients in a prescribed demeanour, protected against an unauthorised entry and transmitted over unreliable channels. All of those operations may be understood purely through an individual with wisdom of fundamentals in algebra and quantity theory.

This booklet offers the mandatory heritage in mathematics, polynomials, teams, fields and elliptic curves that's adequate to appreciate such real-life functions as cryptography, mystery sharing, error-correcting, fingerprinting and compression of data. it's the first to hide many contemporary advancements in those subject matters. according to a lecture path given to third-year undergraduates, it truly is self-contained with a number of labored examples and routines supplied to check knowing. it might probably also be used for self-study.

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Example text

14) are not unique, moreover there exist infinitely many such pairs. However, sometimes, knowing even one pair of such numbers is more important than knowing the greatest common divisor itself. 14) can be easily obtained from the Euclidean algorithm by back substitution. The following theorem provides us with a convenient way of calculating them. It also gives an alternative proof of the existence of m and n based on Linear Algebra. 5 (The Extended Euclidean Algorithm) Let us write the following matrix with two rows R1 , R2 , and three columns C1 , C2 , C3 : [C1 C2 C3 ] = R1 R2 = a10 .

1 Classical Secret-Key Cryptology One of the oldest ciphers known is Atbash. It even appears in the Hebrew Scriptures of the Bible. Any occurrence of the first letter of the alphabet is replaced by the last letter, occurrences of the second letter are replaced by the second to last etc. Atbash is a specific example of a general technique called inversion. Caesar is also a very old cipher used by Gaius Julius Caesar (100 BC–44 BC). Letters are simply replaced by letters three steps further down the alphabet.

CR5 The distributive law holds, that is, a · (b + c) = a · b + a · c, for all a, b, c ∈ R. 4 Other commutative rings include the ring of polynomials Z[x] with integer coefficients or else with rational or real coefficients. The set of all n × n matrices over the integers Zn×n is also a ring but not commutative since axiom CR2 is not satisfied. 3 An element a of a ring R is called invertible if there exists an element b in R such that a · b = b · a = 1. An element b in this case is called a multiplicative inverse of a.

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