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Tổng quan mật mã học Huỳnh Trọng Thưa htthua@ptithcm.edu.vn Introduction • Cryptography was used as a tool to protect national secrets and strategies • 1960s (computers and communications systems) -> means to protect information and to provide security services Introduction (cont.) • 1970s: DES (Feistel, IBM) - the most well-known cryptographic mechanism in history • 1976: public-key cryptography (Diffie and Hellman) • 1978: RSA (Rivest et al.) - first practical public-key encryption and signature scheme • 1991: the first international standard for digital signatures (ISO/IEC 9796) was adopted Information security and cryptography • Some information security objectives – – – – – – – – Privacy or confidentiality Data integrity Entity authentication or identification Message authentication Signature Authorization Validation Access control Information security and cryptography (cont.) • Some information security objectives – – – – – – – – – Certification Timestamping Witnessing Receipt Confirmation Ownership Anonymity non-repudiation Revocation Information security and cryptography (cont.) • Cryptography is the study of mathematical techniques related to aspects of information security such as confidentiality, data integrity, entity authentication, and data origin authentication • Cryptography is not the only means of providing information security, but rather one set of techniques Cryptographic goals • • • • Confidentiality Data integrity Authentication Non-repudiation Cryptography is about the prevention and detection of cheating and other malicious activities A taxonomy of cryptographic primitives Background on functions • Function: f:XY f(x)=y • Ex: f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = f(7) = f(8) = f(9) = f(10) = X = {1, 2, 3, , 10} f(x)= rx, where rx is the remainder when x2 is divided by 11 image of f is the set Y = {1, 3, 4, 5, 9} 1-1 functions • A function is − (injection - đơn ánh) if each element in Y is the image of at most one element in X • A function is onto (toàn ánh) if each element in Y is the image of at least one element in X, i.e Im(f)=Y • If a function f: X → Y is 1−1 and Im(f)=Y, then f is called a bijection (song ánh) 10 Ex (cont.) • One of the major issues with symmetric-key systems is to find an efficient method to agree upon and exchange keys securely -> key distribution problem 27 Block ciphers • A block cipher is an encryption scheme which breaks up the plaintext messages to be transmitted into strings (called blocks) of a fixed length t over an alphabet A, and encrypts one block at a time • Two important classes of block ciphers are substitution ciphers and transposition ciphers 28 Simple substitution ciphers • Let A be an alphabet of q symbols and M be the set of all strings of length t over A • K be the set of all permutations on the set A where m =(m1m2 ···mt) ∈ M • To decrypt c =(c1c2 ··· ct), compute the inverse permutation d = e−1 29 Polyalphabetic substitution ciphers (đa chữ cái) i the key space K consists of all ordered sets of t permutations (p1,p2, ,pt), where each permutation pi is defined on the set A; ii encryption of the message m =(m1m2 ···mt) under the key e =(p1,p2, ,pt) is given by Ee(m)=(p1(m1)p2(m2) ··· pt(mt)); and iii the decryption key associated with e =(p1,p2, ,pt) is d =(p1−1,p2−1, ,pt−1) 30 Ex of Polyalphabetic (Vigenère cipher) • Let A = {A,B,C, ,X,Y, Z} and t =3 Choose e = (p1,p2,p3), where p1 maps each letter to the letter three positions to its right in the alphabet, p2 to the one seven positions to its right, and p3 ten positions to its right If 31 Transposition ciphers (chuyển vị) • Let K be the set of all permutations on the set {1, 2, ,t} For each e ∈ K define the encryption function where m =(m1m2 ···mt) ∈ M • The decryption key corresponding to e is the inverse permutation d = e−1 • To decrypt c =(c1c2 ··· ct), – compute Dd(c)=(cd(1)cd(2) ··· cd(t)) 32 Ex of transposition ciphers e: d = e−1 : Plaintext m: Ciphertext c: 33 Stream ciphers • Let K be the key space, – A sequence of symbols e1e2 ··· ei ∈ K, is called a keystream • Let Ee be a simple substitution cipher with block length where e ∈ K • Let m1m2 ··· be a plaintext string • A stream cipher takes the plaintext string and produces a ciphertext string c1c2 ··· where ci = Eei(mi) – If di denotes the inverse of ei, then Ddi (ci)= mi decrypts the ciphertext string 34 The Vernam cipher • The Vernam Cipher is a stream cipher defined on the alphabet A = {0, 1} • A binary message m1m2 ···mt is operated on by a binary key string k1k2 ··· kt of the same length to produce a ciphertext string c1c2 ··· ct where • If the key string is randomly chosen and never used again, the Vernam cipher is called a one-time pad 35 Digital signatures • M is the set of messages which can be signed • S is a set of elements called signatures, possibly binary strings of a fixed length • SA is a transformation from the message set M to the signature set S, and is called a signing transformation for entity A • The transformation SA is kept secret by A, and will be used to create signatures for messages from M ã VA is a transformation from the set MìS to the set {true, false} – VA is called a verification transformation for A’s signatures, is publicly known, and is used by other entities to verify signatures created by A 36 Ex of digital signature scheme • M= {m1,m2,m3} and S = {s1,s2,s3} 37 Digital signature mechanism • Signing procedure – Compute s = SA(m) – Transmit the pair (m, s) s is called the signature for message m • Verification procedure – Obtain the verification function VA of A – Compute u = VA(m, s) – Accept the signature as having been created by A if u = true, and reject the signature if u = false 38 Public-key cryptography 39 Public-key encryption scheme 40 Hash functions • A hash function is a computationally efficient function mapping binary strings of arbitrary length to binary strings of some fixed length, called hashvalues • It is computationally infeasible to find two distinct inputs which hash to a common value • It is computationally infeasible to find an input (preimage) x such that h(x)= y 41 ... symmetric-key encryption schemes, the term symmetric-key becomes appropriate • Other terms used in the literature are single-key, one-key, private-key, and conventional encryption 25 Ex of symmetric-key... systems) -> means to protect information and to provide security services Introduction (cont.) • 1970s: DES (Feistel, IBM) - the most well-known cryptographic mechanism in history • 1976: public-key... Cryptographic techniques are typically divided into two generic types: symmetric-key and public-key 23 Symmetric-key encryption • Block ciphers • Stream ciphers 24 Overview of block ciphers and