An Introduction to Mathematical Cryptography (2nd Edition) by Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

This self-contained advent to trendy cryptography emphasizes the math in the back of the speculation of public key cryptosystems and electronic signature schemes. The e-book specializes in those key issues whereas constructing the mathematical instruments wanted for the development and defense research of various cryptosystems. merely easy linear algebra is needed of the reader; concepts from algebra, quantity idea, and chance are brought and built as required. this article presents an awesome creation for arithmetic and computing device technological know-how scholars to the mathematical foundations of recent cryptography. The e-book contains an intensive bibliography and index; supplementary fabrics can be found online.

The booklet covers various issues which are thought of valuable to mathematical cryptography. Key issues include:

* classical cryptographic buildings, reminiscent of Diffie–Hellmann key trade, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;

* primary mathematical instruments for cryptography, together with primality checking out, factorization algorithms, chance concept, details idea, and collision algorithms;

* an in-depth therapy of significant cryptographic strategies, akin to elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment variation of An advent to Mathematical Cryptography incorporates a major revision of the cloth on electronic signatures, together with an previous creation to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or increased for readability, specially within the chapters on details thought, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been multiplied to incorporate sections on electronic money and homomorphic encryption. a variety of new workouts were integrated.

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Additional info for An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics)

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It is not surprising that the implementations of inc_l() and dec_l() are similar to those of the functions add_l() and sub_l(). They test for overflow and underflow, respectively, and return the corresponding error codes E_CLINT_OFL and E_CLINT_UFL. Function: increment a CLINT object by 1 Syntax: int inc_l (CLINT a_l); Input: a_l (summand) Output: a_l (sum) Return: E_CLINT_OK if all is ok E_CLINT_OFL if overflow int inc_l (CLINT a_l) { clint *msdptra_l, *aptr_l = LSDPTR_L (a_l); ULONG carry = BASE; int OFL = E_CLINT_OK; msdptra_l = MSDPTR_L (a_l); while ((aptr_l <= msdptra_l) && (carry & BASE)) { *aptr_l = (USHORT)(carry = 1UL + (ULONG)*aptr_l); aptr_l++; } if ((aptr_l > msdptra_l) && (carry & BASE)) { *aptr_l = 1; SETDIGITS_L (a_l, DIGITS_L (a_l) + 1); if (DIGITS_L (a_l) > (USHORT) CLINTMAXDIGIT) /* overflow ?

P0 )B . The following implementation of multiplication contains at its core this main loop. Corresponding to the above estimate, in step 4 the lossless representation of a value less than B 2 in the variable t is required. Analogously to how we proceeded in the case of addition, the inner products t are thus represented as ULONG types. The variable t is nonetheless not used explicitly, and the setting of the result digits pi+j and the carry c occurs rather within a single expression, analogous to the process already mentioned in connection with the addition function (see page 25).

7. Set i ← i + 1; if i ≤ m − 1, go to step 3. 8. Output p = (pm+n−1 pm+n−2 . . p0 )B . The following implementation of multiplication contains at its core this main loop. Corresponding to the above estimate, in step 4 the lossless representation of a value less than B 2 in the variable t is required. Analogously to how we proceeded in the case of addition, the inner products t are thus represented as ULONG types. The variable t is nonetheless not used explicitly, and the setting of the result digits pi+j and the carry c occurs rather within a single expression, analogous to the process already mentioned in connection with the addition function (see page 25).

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