CRYPTOGRAPHIC PROTOCOLS 2015, LECTURE 2
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1 CRYPTOGRAPHIC PROTOCOLS 2015, LECTURE 2 assumptions and reductions Helger Lipmaa University of Tartu, Estonia
2 MOTIVATION Assume Alice designs a protocol How to make sure it is secure? Approach 1: proof by intimidation "don't you trust me" also "tautology" Theorem. Assume Protocol X is secure. Then Protocol X is secure
3 MOTIVATION Approach 2 (much better): prove that the protocol is secure Unconditional security Problem: only known how to do for a small number of protocols need major advances in complexity theory it is not known how to prove that any function takes more than a linear number of steps to compute
4 MOTIVATION Approach 3 (mostly correct): make an assumption: Computational security assume that some well-known problem (say, factoring) is hard prove that if that assumption holds, then the protocol is secure
5 SECURITY VERIFICATION Protocol designer's task Security verifier's task proof by intimidation Simpler: no need to prove anything Spend years cryptanalyzing OR trust the protocol proof by reduction More complex: must reduce security to some assumption Verify the usually short reduction. Trust the assumption
6 SOME KNOWN ASSUMPTIONS Factoring and friends The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. ~~~~~~~ Carl Friedrich Gauss Discrete logarithm and friends Lattice assumptions Many more protocols Coding-theoretic assumptions... Many assumptions
7 SOME ASSUMPTIONS Factoring RSA Strong RSA DCRA Discrete Log CDH DDH Various pairing assumptions SVP RLWE gapsvp CVP LWE
8 SOME ASSUMPTIONS Number-theoretic Z_n Number-theoretic finite cyclic groups Lattice Factoring RSA Strong RSA Discrete Log CDH DDH SVP gapsvp Various pairing assumptions DCRA RLWE Underlying mathematical structure CVP LWE
9 SOME ASSUMPTIONS Factoring RSA Strong RSA Insecure Secure(?) Discrete Log CDH DDH SVP gapsvp Various pairing assumptions DCRA RLWE Security against quantum computers CVP LWE
10 SOME ASSUMPTIONS Strength/familiarity Strong RSA Factoring RSA DCRA Discrete Log CDH DDH Various pairing assumptions More assurance SVP RLWE Often more efficient gapsvp CVP weaker/known LWE stronger/less known
11 ASSUMPTIONS IN THIS COURSE Factoring RSA Strong RSA Discrete Log CDH DDH SVP CVP gapsvp Various pairing assumptions DCRA RLWE LWE no need to know "gray" assumptions
12 CHOICE OF ASSUMPTION: TRADEOFF More security assurance or better efficiency? Quantum security? Number-theoretic flavor? Any algebra needed by the goal? E.g., need to compute a sum of inputs Compatibility with other protocols Some protocols are impossible/very inefficient with some assumptions
13 ASSUMPTION = EVERYTHING Choice of the assumption is very important Seeing an assumption already gives a flavor of what can(not) be done efficiently So let's learn some
14 RECALL: ISOMORPHISM Assume G₁ is additive and G₂ is multiplicative group f : G₁ G₂ is isomorphism if it agrees with group operations f (x + y) = f (x) f (y) f (0) = 1 f (-x) = f (x) ¹ Two groups are isomorphic if there exists isomorphism between them
15 ISOMORPHIC = EQUAL In mathematics, isomorphic groups are considered to be "essentially the same" They have the same structure Instead of executing group operation in one group, you can map to another group, do group operation there, and then map back G₁ + G₁ f f ¹ f can be thought of as data representation G2 G2... assuming both f and f ¹ can be computed efficiently
16 ONE-WAY ISOMORPHISMS f : G₁ G₂ is a one-way isomorphism if 1. f is an isomorphism 2. f can be computed efficiently 3. f ¹ cannot be computed efficiently assumption (not known how to prove such things)
17 ONE-WAY ISOMORPHIC EQUAL G₁ efficient + G₁ efficient f f ¹ not efficient G2 efficient G2
18 RECALL: EXPONENTIATION expg : Zq G, expg (m) = gᵐ expg is isomorphism gᵐ + ⁿ = gᵐ gⁿ g⁰ = 1 g ᵐ = 1 / gᵐ
19 QUIZ: IS EXP ONE-WAY? What do you think? Depends on the group Easy: (R*, ): the inverse of exp is logarithm (Z, +), (Zq, +): exp = multiplication, inverse = division In finite groups, inverse of exp is called discrete logarithm
20 HARD DL GROUPS Instantiation 1 Let p be a big prime (3000+ digits) Best known algorithms to break DL in G have subexponential complexity in p and exponential complexity in q The order of Z p*={1, 2,..., p - 1} is p - 1 Let q (p - 1) be a smaller prime (160+ digits) By Cauchy/Sylow theorem, Zp* has a unique subgroup G of order q DL is assumed to be hard in G
21 REMINDER: BASIC COMPLEXITY THEORY Running time T(n) of algorithm = function of input length n Example. Running time of exponentiation is a function of the bitlength n of the group elements Simplification: in Zq, n = log q Efficient algorithm: T(n) is polynomial in n E.g.: T(n) = 1000 n^6 Inefficient algorithm: T(n) is not polynomial in n E.g.: T(n) = n^(log n)
22 EFFICIENT VS INEFFICIENT... but it will eventually become less efficient Superpolynomial might be more efficient for small n
23 COMPLEXITY IN CRYPTOGRAPHY When we encrypt, security should not depend on the message length but say on key size Instead of input length n, take security parameter κ Usually κ related to key length First, fix κ so that T(κ) of attacks is big and of "honest" algorithms is small Finally, choose corresponding key
24 COROLLARIES OF COMPLEXITY Most algorithms work with undetermined κ In practical implementations fix κ so that protocol is fast but attacks are assumed to be hard E.g., attacks take time 2⁸⁰ If attacks are improved somewhat, increase κ accordingly
25 CHOOSING P p Instantiation 1 log of time Efficient protocol has small value here Κ(p) time of index calculus: best known DL (appr) time of exponentiation (appr) log p Too optimistic graph, in practice p is much larger
26 COMPLEXITY NOTATION Θ (f (n)): asymptotically c f (n) for some constant c 100 n²+ 20 n - 10 = Θ (n²) O (f (n)): any func that does not grow faster than Θ (f (n)) o (f (n)): any function that grows slower than Θ (f (n)) Ω (f (n)): any func that does not grow slower than Θ (f (n)) ω (f (n)): any function that grows faster than Θ (f (n))
27 QUIZ Θ (n⁸) O (n⁸) Ω (n⁸) o (n⁸) ω (n⁸) Θ (n⁷) O (n⁷) Ω (n⁷) o (n⁷) ω (n⁷) Θ (n⁶) O (n⁶) Ω (n⁶) o (n⁶) ω (n⁶) Θ (n⁵) O (n⁵) Ω (n⁵) o (n⁵) ω (n⁵) Question: What is (n⁸ + n + 1) / (n² + n + 1)? Answer: it is n⁶ + smaller terms thus Θ (n⁶)
28 COMPLEXITY NOTATION polynomial: poly (n) = n^(o (1)) not faster than any polynomial superpolynomial: n^(ω (1)) faster than any polynomial exponential: 2^(Θ (n)) negligible: negl (n) = n^(-ω (1)) slower than inverse of any polynomial linear: Θ (n) asymptotically c n for some constant c etc: logarithmic, superlogarithmic, sublinear
29 BEST KNOWN DL ALGORITHMS Any groups of order q, n := log q Baby-step-giant-step and Pohlig-Hellman algorithms --- O ( q) Instantiation 1, parameters p and q Index calculus, O (e^( (2 ln p ln ln p))) Generic algorithms: only use group operations BSGS/PH algorithms O ( q) Recent advances in groups of order pᵐ for midsize m DL in any group can be broken by using quantum computer
30 HARD DL GROUPS Instantiation 2 Elliptic curve groups Let q be a small prime (160+ digits) Elliptic curve group G has order q Definition complicated (see supplementary notes) Best known algorithms to break DL in G have exponential complexity in q DL is assumed to be hard in well-chosen G
31 COMPARISON OF INSTANTIATIONS Exponent 1.58 due to Karatsuba algorithm Asymptotically not optimal, but good for inputs of that size Parameters Group element representation Complexity of multiplication Security Zp* p, log p 3200 q, log q 160 log p O((log p)^1.58) 2⁸⁰ E.C.G. q, log q 160 log q O((log q)^1.58) 2⁸⁰ q is much smaller than p, though constant in O( ) is larger
32 QUIZ Question: What is the asymptotic efficiency of exponentiation in the first instantiation? q = 2^(Θ (κ)), thus log q = Θ (κ) 2^κ = Θ (e^ (ln p ln ln p))) thus log p = Θ (κ² / log κ) Answer: complexity of multiplication times Θ (log p) multiplications Θ ((log p)^2.58) = Θ (κ^5.16 / (log κ)^2.58) Θ ((log p)^1.58) O ((log p)^2.58) O ((log p)^3) Ω ((log q)^1.58) O ((log q)^2.58) o ((log q)⁶) O ((log p)^1.58 log q) o ((log p)^2.58) O ((log q)^1.58 log p) Ω (1) Θ ((log q)^5.16) ω ((log p)^1.58)
33 DL ASSUMPTION: FORMAL Informally, we need that inverting exponentiation is hard Complications: when exponent is smaller than L, one can compute DL in Θ ( L) steps inverting is impossible when g = 1 inverting is always possible with probability 1 / q (guessing answer randomly) exponent must be random (e.g., exponent is secret key) g must be a generator security must hold against probabilistic algorithms that can use random numbers break is only successful when adversary's advantage is >> 1 / q
34 SECURITY GAME A challenger generates values from some fixed "valid" distributions and sends them to the adversary A Depending on the input and the output, the challenger declares A to be either successful or not After some computation, A returns some value to the challenger A breaks the assumption if her advantage is big compared to random guessing
35 DEF: DL GROUPS Let G be a finite cyclic group of order q, let g be its fixed generator One can take any g, or a random g Assume desc(g) contains a description of G, incl. g Adv[DL(G, A)] := Pr[DL(G,A) = 1] - 1 / q A ε-breaks DL in G iff Adv[DL(G, A)] ε G is a (τ,ε)-dl group iff Adv[DL(G, A)] ε for all probabilistic polynomial time adversaries A that take time τ G is a DL group iff it is a (poly(κ),negl(κ))-dl group Game DL(G, A) gk desc(g) m Zq h gᵐ m* A (gk, h) If m = m* return 1 else return 0
36 STUDY OUTCOMES Assumptions: motivation Some example assumptions, why so many Discrete logarithm basic idea formal definition
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