Download An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman PDF

By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

An advent to Mathematical Cryptography offers an creation to public key cryptography and underlying arithmetic that's required for the topic. all the 8 chapters expands on a particular region of mathematical cryptography and gives an intensive checklist of exercises.

It is an appropriate textual content for complicated scholars in natural and utilized arithmetic and computing device technological know-how, or the ebook can be used as a self-study. This ebook additionally presents a self-contained therapy of mathematical cryptography for the reader with restricted mathematical background.

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Extra resources for An Introduction to Mathematical Cryptography

Example text

Since each number in the sequence is the square of the preceding one. Further, since we only need these values modulo 1000, we never need to store more 7 than three digits. 8 lists the powers of 3 modulo 1000 up to 32 . 8 requires only 7 multiplications, despite the fact that the 7 number 32 = 3128 has quite a large exponent, because each successive entry in the table is equal to the square of the previous entry. 8 are needed to compute 3218 . 3. 8: Successive square powers of 3 modulo 1000 3 4 6 3218 = 32 · 32 · 32 · 32 · 32 7 ≡ 9 · 561 · 721 · 281 · 961 ≡ 489 (mod 1000).

Finally, we substitute the expressions 220 = −a + 3b and 88 = 3a − 8b into the penultimate line to get −a + 3b = (3a − 8b) · 2 + 44, so 44 = −7a + 19b. In other words, −7 · 2024 + 19 · 748 = 44 = gcd(2024, 748), so we have found a way to write gcd(a, b) as a linear combination of a and b using integer coefficients. In general, it is always possible to write gcd(a, b) as an integer linear combination of a and b, a simple sounding result with many important consequences. 11 (Extended Euclidean Algorithm).

Decryption The fast powering algorithm In some cryptosystems that we will study, for example the RSA and Diffie– Hellman cryptosystems, Alice and Bob are required to compute large powers of a number g modulo another number N , where N may have hundreds of digits. The naive way to compute g A is by repeated multiplication by g. Thus g1 ≡ g (mod N ), g2 ≡ g · g1 (mod N ), g3 ≡ g · g2 (mod N ), g4 ≡ g · g3 (mod N ), g5 ≡ g · g4 (mod N ), . . It is clear that gA ≡ g A (mod N ), but if A is large, this algorithm is completely impractical.

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