### Table 12.1: Pseudorandom generators of flelded stream ciphers

2004

### Table 2: Overview of pseudo-random generator constructions.

in Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses

"... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table 2 for derandomizing the process. The symbol A... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table2 for derandomizing the process. The symbol A... In PAGE 3: ...Table2... In PAGE 8: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table 2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 8: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table 2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 9: ...More Applications and New Derandomizations We will now apply the general framework of Section 4 to various fundamental constructions in computational complexity. As customary, we only state our results in terms of the strongest of the assumptions in Table2 , yielding polynomial time deterministic simulations. It should be noted, however, that weaker assumptions can be taken (the weakest being that the polynomial- time hierarchy does not collapse) in order to achieve weaker, but still subexponential, deterministic simulations.... ..."

### Table 2: Overview of pseudo-random generator constructions.

in Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses

1999

"... In PAGE 2: ... We formally define the notion of a success pred- icate in Section 4. If we can decide the success predicate of a ran- domized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table2 provides a pseudo-random generator G with the characteristics on the right- hand side of Table 2 for derandomizing the process. The symbol A in Table 2 represents an arbitrary class of oracles.... In PAGE 2: ... We formally define the notion of a success pred- icate in Section 4. If we can decide the success predicate of a ran- domized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right- hand side of Table2 for derandomizing the process. The symbol A in Table 2 represents an arbitrary class of oracles.... In PAGE 2: ... If we can decide the success predicate of a ran- domized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right- hand side of Table 2 for derandomizing the process. The symbol A in Table2 represents an arbitrary class of oracles. To illustrate the power of our generalization, we apply it to the following randomized processes from different areas of theoretical computer science.... In PAGE 5: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of ran- dom bits, and suppose that B can efficiently check (F; ). Then the hardness conditions on the left-hand side of Table2 provide pseudo-random generators G with complexity and seed lengths s as specified on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x;Gnd( )) = 1]j 2 o(1): The parameter s in Table 2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will first analyze the complexity of an oracle B capable of effi- ciently checking the associated success predicate and then con- struct a pseudo-random generator secure against B based on a func- tion with presumed hardness against B.... In PAGE 5: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of ran- dom bits, and suppose that B can efficiently check (F; ). Then the hardness conditions on the left-hand side of Table 2 provide pseudo-random generators G with complexity and seed lengths s as specified on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x;Gnd( )) = 1]j 2 o(1): The parameter s in Table2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will first analyze the complexity of an oracle B capable of effi- ciently checking the associated success predicate and then con- struct a pseudo-random generator secure against B based on a func- tion with presumed hardness against B.... In PAGE 5: ... 5 More Applications We will now apply the general framework of Section 4 to various other constructions in computational complexity. As customary, we only state our results in terms of the strongest of the assumptions in Table2 , yielding polynomial time deterministic simulations. It should be noted, however, that weaker assumptions can be taken (e.... ..."

Cited by 72

### Table 2. VMPC stream cipher

"... In PAGE 8: ...2. Execute steps 1 and 2 of the VMPC stream cipher ( Table2 ) and save the 20 generated outputs as a 160-bit MAC 7 Test values of the VMPC-Tail-MAC scheme Table 5 gives an example 20-byte tag generated by the VMPC-Tail-MAC scheme for a given 16-byte key (K), a given 16-byte Initialization Vector (V ) and for a 256-byte plaintext Message consisting of consecutive numbers from 0 to 255... ..."

Cited by 1

### Table 2. VMPC stream cipher

"... In PAGE 8: ...2. Execute steps 1 and 2 of the VMPC stream cipher ( Table2 ) and save the 20 generated outputs as a 160-bit MAC 7 Test values of the VMPC-Tail-MAC scheme Table 5 gives an example 20-byte tag generated by the VMPC-Tail-MAC scheme for a given 16-byte key (K), a given 16-byte Initialization Vector (V ) and for a 256-byte plaintext Message consisting of consecutive numbers from 0 to 255... ..."

Cited by 1

### Table 2. VMPC stream cipher

"... In PAGE 8: ...2. Execute steps 1 and 2 of the VMPC stream cipher ( Table2 ) and save the 20 generated outputs as a 160-bit MAC 7 Test values of the VMPC-Tail-MAC scheme Table 5 gives an example 20-byte tag generated by the VMPC-Tail-MAC scheme for a given 16-byte key (K), a given 16-byte Initialization Vector (V ) and for a 256-byte plaintext Message consisting of consecutive numbers from 0 to 255... ..."

### Table 4. New York State Distributed Generation Resources: Number of Generators

in NOTICE

"... In PAGE 27: ...ounty. There are more than 3,582 MW of capacity across the state. Most of this capacity, 2,896 MW or 80%, is from reciprocating engines, and the balance is from turbines. Table4 shows the number of generators by type and by county. There were 10,542 generators of at least 100 kW of capacity in all of New York State in the database.... In PAGE 77: ...ct 19.05 148.46 47.91 Table 11 shows the hours of self-generation bid in the DAM. As shown in Table4 , August had the most curtailment hours, with each of the three blocks operated at least 16 hours. In July, there were five days when some or all of the sites provided self-generation: July 2, 3, 16, 18, and 23.... ..."

### Table 4: Instances arising from pseudo-randomly distributed generators 24

1994

"... In PAGE 23: ... If we are not in the \nice case quot;, both methods have to make a similar number of comparisons, and therefore, if the \bad cases quot; outweigh, the di erence of e ciency of the two methods gets smaller. Table4 shows the same running time data for generator sets being uniformly pseudo- randomly distributed in [0; 1]2. The rst column contains the number of generators.... ..."

Cited by 2

### Table 2: Overview of pseudo-random generator constructions. To illustrate the power of our generalization, we apply our technique to the following fundamental constructions from di erent areas of theoretical computer science. 2

in Graph Nonisomorphism Has Subexponential Size Proofs Unless The Polynomial-Time Hierarchy Collapses

"... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table 2 for derandomizing the process. The symbol A... In PAGE 3: ... We formally de ne the notion of a success predicate in Section 4. If we can decide the success predicate of a randomized process with polynomial size B-oracle circuits, then the hardness assumption on the left-hand side of Table 2 provides a pseudo-random generator G with the characteristics on the right-hand side of Table2 for derandomizing the process. The symbol A... In PAGE 3: ...Table2... In PAGE 9: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table 2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 9: ...heorem 4.2 Let A be a class of oracles and B an oracle. Let (F; ) be a randomized process using a polynomial number of random bits, and suppose that B can e ciently check (F; ). Then the hardness conditions of the left-hand side of Table 2 provide a pseudo-random generator G with complexity and seed length s as speci ed on the right-hand side of the table such that for some constant d gt; 0 and any input x of length n j Pr [ (x; ) = 1] ? Pr [ (x; Gnd( )) = 1]j 2 o(1): The parameter s in Table2 can be any space constructible function. In order to reduce the randomness of a randomized process, we will rst analyze the complexity of an oracle B capable of e ciently checking the associated success predicate and then construct a pseudo-random generator secure against B based on a function with presumed hardness against B.... In PAGE 9: ... 5 More Applications and New Derandomizations We will now apply the general framework of Section 4 to various fundamental constructions in computational complexity. As customary, we only state our results in terms of the strongest of the assumptions in Table2 , yielding polynomial time deterministic simulations. It should be noted, how- ever, that weaker assumptions can be taken (the weakest being that the polynomial-time hierarchy does not collapse) in order to achieve weaker, but still subexponential, deterministic simulations.... ..."

### Table 1. Comparison of randomness of several schemes (when d =2;k=4;m= 128)

1999

"... In PAGE 13: ... 4 Conclusion We studied randomness provided by several schemes used for block ciphers. We focused on the schemes for AES candidates, in particular (see Table1... In PAGE 14: ... We also assumed that internal primitives are ideal random ones. The results in Table1 show that the Feistel scheme is the best in that the required number of rounds for pseudorandomness and super-pseudorandomness is the smallest. However, in comparing the randomness among several schemes we should take account of the computational cost of random primitives.... ..."

Cited by 3